Find the Degree, Leading Term, and Leading Coefficient (x+1)^3(x^2-1)^2(1-x^4)(2x+3)

${(x + 1)}^{3} {(x^{2} - 1)}^{2} (1 - x^{4}) (2 x + 3)$

Step 1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

Use the Binomial Theorem.

$(x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3}) {(x^{2} - 1)}^{2} (1 - x^{4}) (2 x + 3)$

Simplify terms.

Simplify each term.

Multiply $3$ by $1$ .

$(x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3}) {(x^{2} - 1)}^{2} (1 - x^{4}) (2 x + 3)$

One to any power is one.

$(x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3}) {(x^{2} - 1)}^{2} (1 - x^{4}) (2 x + 3)$

Multiply $3$ by $1$ .

$(x^{3} + 3 x^{2} + 3 x + 1^{3}) {(x^{2} - 1)}^{2} (1 - x^{4}) (2 x + 3)$

One to any power is one.

$(x^{3} + 3 x^{2} + 3 x + 1) {(x^{2} - 1)}^{2} (1 - x^{4}) (2 x + 3)$

Rewrite ${(x^{2} - 1)}^{2}$ as $(x^{2} - 1) (x^{2} - 1)$ .

$(x^{3} + 3 x^{2} + 3 x + 1) ((x^{2} - 1) (x^{2} - 1)) (1 - x^{4}) (2 x + 3)$

Expand $(x^{2} - 1) (x^{2} - 1)$ using the FOIL Method.

Apply the distributive property.

$(x^{3} + 3 x^{2} + 3 x + 1) (x^{2} (x^{2} - 1) - 1 (x^{2} - 1)) (1 - x^{4}) (2 x + 3)$

Apply the distributive property.

$(x^{3} + 3 x^{2} + 3 x + 1) (x^{2} x^{2} + x^{2} \cdot - 1 - 1 (x^{2} - 1)) (1 - x^{4}) (2 x + 3)$

Apply the distributive property.

$(x^{3} + 3 x^{2} + 3 x + 1) (x^{2} x^{2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1) (1 - x^{4}) (2 x + 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{3} + 3 x^{2} + 3 x + 1) (x^{2 + 2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1) (1 - x^{4}) (2 x + 3)$

Add $2$ and $2$ .

$(x^{3} + 3 x^{2} + 3 x + 1) (x^{4} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1) (1 - x^{4}) (2 x + 3)$

Move $- 1$ to the left of $x^{2}$ .

$(x^{3} + 3 x^{2} + 3 x + 1) (x^{4} - 1 \cdot x^{2} - 1 x^{2} - 1 \cdot - 1) (1 - x^{4}) (2 x + 3)$

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$(x^{3} + 3 x^{2} + 3 x + 1) (x^{4} - x^{2} - 1 x^{2} - 1 \cdot - 1) (1 - x^{4}) (2 x + 3)$

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$(x^{3} + 3 x^{2} + 3 x + 1) (x^{4} - x^{2} - x^{2} - 1 \cdot - 1) (1 - x^{4}) (2 x + 3)$

Multiply $- 1$ by $- 1$ .

$(x^{3} + 3 x^{2} + 3 x + 1) (x^{4} - x^{2} - x^{2} + 1) (1 - x^{4}) (2 x + 3)$

Subtract $x^{2}$ from $- x^{2}$ .

$(x^{3} + 3 x^{2} + 3 x + 1) (x^{4} - 2 x^{2} + 1) (1 - x^{4}) (2 x + 3)$

Expand $(x^{3} + 3 x^{2} + 3 x + 1) (x^{4} - 2 x^{2} + 1)$ by multiplying each term in the first expression by each term in the second expression.

$(x^{3} x^{4} + x^{3} (- 2 x^{2}) + x^{3} \cdot 1 + 3 x^{2} x^{4} + 3 x^{2} (- 2 x^{2}) + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Simplify terms.

Simplify each term.

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{3 + 4} + x^{3} (- 2 x^{2}) + x^{3} \cdot 1 + 3 x^{2} x^{4} + 3 x^{2} (- 2 x^{2}) + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Add $3$ and $4$ .

$(x^{7} + x^{3} (- 2 x^{2}) + x^{3} \cdot 1 + 3 x^{2} x^{4} + 3 x^{2} (- 2 x^{2}) + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Rewrite using the commutative property of multiplication.

$(x^{7} - 2 x^{3} x^{2} + x^{3} \cdot 1 + 3 x^{2} x^{4} + 3 x^{2} (- 2 x^{2}) + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(x^{7} - 2 (x^{2} x^{3}) + x^{3} \cdot 1 + 3 x^{2} x^{4} + 3 x^{2} (- 2 x^{2}) + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 2 x^{2 + 3} + x^{3} \cdot 1 + 3 x^{2} x^{4} + 3 x^{2} (- 2 x^{2}) + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Add $2$ and $3$ .

$(x^{7} - 2 x^{5} + x^{3} \cdot 1 + 3 x^{2} x^{4} + 3 x^{2} (- 2 x^{2}) + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $x^{3}$ by $1$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{2} x^{4} + 3 x^{2} (- 2 x^{2}) + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 (x^{4} x^{2}) + 3 x^{2} (- 2 x^{2}) + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{4 + 2} + 3 x^{2} (- 2 x^{2}) + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Add $4$ and $2$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} + 3 x^{2} (- 2 x^{2}) + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Rewrite using the commutative property of multiplication.

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} + 3 \cdot - 2 x^{2} x^{2} + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} + 3 \cdot - 2 (x^{2} x^{2}) + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} + 3 \cdot - 2 x^{2 + 2} + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Add $2$ and $2$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} + 3 \cdot - 2 x^{4} + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $3$ by $- 2$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} \cdot 1 + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $3$ by $1$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x \cdot x^{4} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $x$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 (x^{4} x) + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $x^{4}$ by $x$ .

Raise $x$ to the power of $1$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 (x^{4} x^{1}) + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x^{4 + 1} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Add $4$ and $1$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x^{5} + 3 x (- 2 x^{2}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Rewrite using the commutative property of multiplication.

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x^{5} + 3 \cdot - 2 x \cdot x^{2} + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x^{5} + 3 \cdot - 2 (x^{2} x) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x^{5} + 3 \cdot - 2 (x^{2} x^{1}) + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x^{5} + 3 \cdot - 2 x^{2 + 1} + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Add $2$ and $1$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x^{5} + 3 \cdot - 2 x^{3} + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $3$ by $- 2$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x^{5} - 6 x^{3} + 3 x \cdot 1 + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $3$ by $1$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x^{5} - 6 x^{3} + 3 x + 1 x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $x^{4}$ by $1$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x^{5} - 6 x^{3} + 3 x + x^{4} + 1 (- 2 x^{2}) + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $- 2 x^{2}$ by $1$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x^{5} - 6 x^{3} + 3 x + x^{4} - 2 x^{2} + 1 \cdot 1) (1 - x^{4}) (2 x + 3)$

Multiply $1$ by $1$ .

$(x^{7} - 2 x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} + 3 x^{5} - 6 x^{3} + 3 x + x^{4} - 2 x^{2} + 1) (1 - x^{4}) (2 x + 3)$

Simplify by adding terms.

Add $- 2 x^{5}$ and $3 x^{5}$ .

$(x^{7} + x^{5} + x^{3} + 3 x^{6} - 6 x^{4} + 3 x^{2} - 6 x^{3} + 3 x + x^{4} - 2 x^{2} + 1) (1 - x^{4}) (2 x + 3)$

Subtract $6 x^{3}$ from $x^{3}$ .

$(x^{7} + x^{5} + 3 x^{6} - 6 x^{4} + 3 x^{2} - 5 x^{3} + 3 x + x^{4} - 2 x^{2} + 1) (1 - x^{4}) (2 x + 3)$

Add $- 6 x^{4}$ and $x^{4}$ .

$(x^{7} + x^{5} + 3 x^{6} - 5 x^{4} + 3 x^{2} - 5 x^{3} + 3 x - 2 x^{2} + 1) (1 - x^{4}) (2 x + 3)$

Subtract $2 x^{2}$ from $3 x^{2}$ .

$(x^{7} + x^{5} + 3 x^{6} - 5 x^{4} + x^{2} - 5 x^{3} + 3 x + 1) (1 - x^{4}) (2 x + 3)$

Expand $(x^{7} + x^{5} + 3 x^{6} - 5 x^{4} + x^{2} - 5 x^{3} + 3 x + 1) (1 - x^{4})$ by multiplying each term in the first expression by each term in the second expression.

$(x^{7} \cdot 1 + x^{7} (- x^{4}) + x^{5} \cdot 1 + x^{5} (- x^{4}) + 3 x^{6} \cdot 1 + 3 x^{6} (- x^{4}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Simplify terms.

Simplify each term.

Multiply $x^{7}$ by $1$ .

$(x^{7} + x^{7} (- x^{4}) + x^{5} \cdot 1 + x^{5} (- x^{4}) + 3 x^{6} \cdot 1 + 3 x^{6} (- x^{4}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Rewrite using the commutative property of multiplication.

$(x^{7} - x^{7} x^{4} + x^{5} \cdot 1 + x^{5} (- x^{4}) + 3 x^{6} \cdot 1 + 3 x^{6} (- x^{4}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $x^{7}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$(x^{7} - (x^{4} x^{7}) + x^{5} \cdot 1 + x^{5} (- x^{4}) + 3 x^{6} \cdot 1 + 3 x^{6} (- x^{4}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - x^{4 + 7} + x^{5} \cdot 1 + x^{5} (- x^{4}) + 3 x^{6} \cdot 1 + 3 x^{6} (- x^{4}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Add $4$ and $7$ .

$(x^{7} - x^{11} + x^{5} \cdot 1 + x^{5} (- x^{4}) + 3 x^{6} \cdot 1 + 3 x^{6} (- x^{4}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $x^{5}$ by $1$ .

$(x^{7} - x^{11} + x^{5} + x^{5} (- x^{4}) + 3 x^{6} \cdot 1 + 3 x^{6} (- x^{4}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Rewrite using the commutative property of multiplication.

$(x^{7} - x^{11} + x^{5} - x^{5} x^{4} + 3 x^{6} \cdot 1 + 3 x^{6} (- x^{4}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$(x^{7} - x^{11} + x^{5} - (x^{4} x^{5}) + 3 x^{6} \cdot 1 + 3 x^{6} (- x^{4}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - x^{11} + x^{5} - x^{4 + 5} + 3 x^{6} \cdot 1 + 3 x^{6} (- x^{4}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Add $4$ and $5$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} \cdot 1 + 3 x^{6} (- x^{4}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $3$ by $1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} + 3 x^{6} (- x^{4}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Rewrite using the commutative property of multiplication.

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} + 3 \cdot - 1 x^{6} x^{4} - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $x^{6}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} + 3 \cdot - 1 (x^{4} x^{6}) - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} + 3 \cdot - 1 x^{4 + 6} - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Add $4$ and $6$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} + 3 \cdot - 1 x^{10} - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $3$ by $- 1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} \cdot 1 - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $- 5$ by $1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} - 5 x^{4} (- x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Rewrite using the commutative property of multiplication.

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} - 5 \cdot - 1 x^{4} x^{4} + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $x^{4}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} - 5 \cdot - 1 (x^{4} x^{4}) + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} - 5 \cdot - 1 x^{4 + 4} + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Add $4$ and $4$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} - 5 \cdot - 1 x^{8} + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $- 5$ by $- 1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} \cdot 1 + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $x^{2}$ by $1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} + x^{2} (- x^{4}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Rewrite using the commutative property of multiplication.

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{2} x^{4} - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - (x^{4} x^{2}) - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{4 + 2} - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Add $4$ and $2$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} \cdot 1 - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $- 5$ by $1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} - 5 x^{3} (- x^{4}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Rewrite using the commutative property of multiplication.

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} - 5 \cdot - 1 x^{3} x^{4} + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} - 5 \cdot - 1 (x^{4} x^{3}) + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} - 5 \cdot - 1 x^{4 + 3} + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Add $4$ and $3$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} - 5 \cdot - 1 x^{7} + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $- 5$ by $- 1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} + 5 x^{7} + 3 x \cdot 1 + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $3$ by $1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} + 5 x^{7} + 3 x + 3 x (- x^{4}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Rewrite using the commutative property of multiplication.

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} + 5 x^{7} + 3 x + 3 \cdot - 1 x \cdot x^{4} + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $x$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} + 5 x^{7} + 3 x + 3 \cdot - 1 (x^{4} x) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $x^{4}$ by $x$ .

Raise $x$ to the power of $1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} + 5 x^{7} + 3 x + 3 \cdot - 1 (x^{4} x^{1}) + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} + 5 x^{7} + 3 x + 3 \cdot - 1 x^{4 + 1} + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Add $4$ and $1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} + 5 x^{7} + 3 x + 3 \cdot - 1 x^{5} + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $3$ by $- 1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} + 5 x^{7} + 3 x - 3 x^{5} + 1 \cdot 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $1$ by $1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} + 5 x^{7} + 3 x - 3 x^{5} + 1 + 1 (- x^{4})) (2 x + 3)$

Multiply $- x^{4}$ by $1$ .

$(x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} + 5 x^{7} + 3 x - 3 x^{5} + 1 - x^{4}) (2 x + 3)$

Simplify by adding terms.

Add $x^{7}$ and $5 x^{7}$ .

$(6 x^{7} - x^{11} + x^{5} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} + 3 x - 3 x^{5} + 1 - x^{4}) (2 x + 3)$

Subtract $3 x^{5}$ from $x^{5}$ .

$(6 x^{7} - x^{11} - x^{9} + 3 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - x^{6} - 5 x^{3} + 3 x - 2 x^{5} + 1 - x^{4}) (2 x + 3)$

Subtract $x^{6}$ from $3 x^{6}$ .

$(6 x^{7} - x^{11} - x^{9} + 2 x^{6} - 3 x^{10} - 5 x^{4} + 5 x^{8} + x^{2} - 5 x^{3} + 3 x - 2 x^{5} + 1 - x^{4}) (2 x + 3)$

Subtract $x^{4}$ from $- 5 x^{4}$ .

$(6 x^{7} - x^{11} - x^{9} + 2 x^{6} - 3 x^{10} - 6 x^{4} + 5 x^{8} + x^{2} - 5 x^{3} + 3 x - 2 x^{5} + 1) (2 x + 3)$

Expand $(6 x^{7} - x^{11} - x^{9} + 2 x^{6} - 3 x^{10} - 6 x^{4} + 5 x^{8} + x^{2} - 5 x^{3} + 3 x - 2 x^{5} + 1) (2 x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$6 x^{7} (2 x) + 6 x^{7} \cdot 3 - x^{11} (2 x) - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$6 \cdot 2 x^{7} x + 6 x^{7} \cdot 3 - x^{11} (2 x) - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x^{7}$ by $x$ by adding the exponents.

Move $x$ .

$6 \cdot 2 (x \cdot x^{7}) + 6 x^{7} \cdot 3 - x^{11} (2 x) - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x$ by $x^{7}$ .

Raise $x$ to the power of $1$ .

$6 \cdot 2 (x^{1} x^{7}) + 6 x^{7} \cdot 3 - x^{11} (2 x) - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$6 \cdot 2 x^{1 + 7} + 6 x^{7} \cdot 3 - x^{11} (2 x) - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Add $1$ and $7$ .

$6 \cdot 2 x^{8} + 6 x^{7} \cdot 3 - x^{11} (2 x) - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $6$ by $2$ .

$12 x^{8} + 6 x^{7} \cdot 3 - x^{11} (2 x) - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $3$ by $6$ .

$12 x^{8} + 18 x^{7} - x^{11} (2 x) - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Rewrite using the commutative property of multiplication.

$12 x^{8} + 18 x^{7} - 1 \cdot 2 x^{11} x - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x^{11}$ by $x$ by adding the exponents.

Move $x$ .

$12 x^{8} + 18 x^{7} - 1 \cdot 2 (x \cdot x^{11}) - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x$ by $x^{11}$ .

Raise $x$ to the power of $1$ .

$12 x^{8} + 18 x^{7} - 1 \cdot 2 (x^{1} x^{11}) - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{8} + 18 x^{7} - 1 \cdot 2 x^{1 + 11} - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Add $1$ and $11$ .

$12 x^{8} + 18 x^{7} - 1 \cdot 2 x^{12} - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $- 1$ by $2$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - x^{11} \cdot 3 - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $3$ by $- 1$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - x^{9} (2 x) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Rewrite using the commutative property of multiplication.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 1 \cdot 2 x^{9} x - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x^{9}$ by $x$ by adding the exponents.

Move $x$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 1 \cdot 2 (x \cdot x^{9}) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x$ by $x^{9}$ .

Raise $x$ to the power of $1$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 1 \cdot 2 (x^{1} x^{9}) - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 1 \cdot 2 x^{1 + 9} - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Add $1$ and $9$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 1 \cdot 2 x^{10} - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $- 1$ by $2$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - x^{9} \cdot 3 + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $3$ by $- 1$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 2 x^{6} (2 x) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Rewrite using the commutative property of multiplication.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 2 \cdot 2 x^{6} x + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x^{6}$ by $x$ by adding the exponents.

Move $x$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 2 \cdot 2 (x \cdot x^{6}) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x$ by $x^{6}$ .

Raise $x$ to the power of $1$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 2 \cdot 2 (x^{1} x^{6}) + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 2 \cdot 2 x^{1 + 6} + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Add $1$ and $6$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 2 \cdot 2 x^{7} + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $2$ by $2$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 2 x^{6} \cdot 3 - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $3$ by $2$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 3 x^{10} (2 x) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Rewrite using the commutative property of multiplication.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 3 \cdot 2 x^{10} x - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x^{10}$ by $x$ by adding the exponents.

Move $x$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 3 \cdot 2 (x \cdot x^{10}) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x$ by $x^{10}$ .

Raise $x$ to the power of $1$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 3 \cdot 2 (x^{1} x^{10}) - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 3 \cdot 2 x^{1 + 10} - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Add $1$ and $10$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 3 \cdot 2 x^{11} - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $- 3$ by $2$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 3 x^{10} \cdot 3 - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $3$ by $- 3$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 6 x^{4} (2 x) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Rewrite using the commutative property of multiplication.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 6 \cdot 2 x^{4} x - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 6 \cdot 2 (x \cdot x^{4}) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x$ by $x^{4}$ .

Raise $x$ to the power of $1$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 6 \cdot 2 (x^{1} x^{4}) - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 6 \cdot 2 x^{1 + 4} - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Add $1$ and $4$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 6 \cdot 2 x^{5} - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $- 6$ by $2$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 6 x^{4} \cdot 3 + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $3$ by $- 6$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 5 x^{8} (2 x) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Rewrite using the commutative property of multiplication.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 5 \cdot 2 x^{8} x + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x^{8}$ by $x$ by adding the exponents.

Move $x$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 5 \cdot 2 (x \cdot x^{8}) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x$ by $x^{8}$ .

Raise $x$ to the power of $1$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 5 \cdot 2 (x^{1} x^{8}) + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 5 \cdot 2 x^{1 + 8} + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Add $1$ and $8$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 5 \cdot 2 x^{9} + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $5$ by $2$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 5 x^{8} \cdot 3 + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $3$ by $5$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + x^{2} (2 x) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Rewrite using the commutative property of multiplication.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{2} x + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 (x \cdot x^{2}) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 (x^{1} x^{2}) + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{1 + 2} + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Add $1$ and $2$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + x^{2} \cdot 3 - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Move $3$ to the left of $x^{2}$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 \cdot x^{2} - 5 x^{3} (2 x) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Rewrite using the commutative property of multiplication.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 5 \cdot 2 x^{3} x - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 5 \cdot 2 (x \cdot x^{3}) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x$ by $x^{3}$ .

Raise $x$ to the power of $1$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 5 \cdot 2 (x^{1} x^{3}) - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 5 \cdot 2 x^{1 + 3} - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Add $1$ and $3$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 5 \cdot 2 x^{4} - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $- 5$ by $2$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 5 x^{3} \cdot 3 + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $3$ by $- 5$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 3 x (2 x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Rewrite using the commutative property of multiplication.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 3 \cdot 2 x \cdot x + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 3 \cdot 2 (x \cdot x) + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x$ by $x$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 3 \cdot 2 x^{2} + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $3$ by $2$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 3 x \cdot 3 - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $3$ by $3$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x - 2 x^{5} (2 x) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Rewrite using the commutative property of multiplication.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x - 2 \cdot 2 x^{5} x - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x^{5}$ by $x$ by adding the exponents.

Move $x$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x - 2 \cdot 2 (x \cdot x^{5}) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $x$ by $x^{5}$ .

Raise $x$ to the power of $1$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x - 2 \cdot 2 (x^{1} x^{5}) - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x - 2 \cdot 2 x^{1 + 5} - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Add $1$ and $5$ .

$12 x^{8} + 18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 15 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x - 2 \cdot 2 x^{6} - 2 x^{5} \cdot 3 + 1 (2 x) + 1 \cdot 3$

Multiply $- 2$ by $2$ .

Multiply $3$ by $- 2$ .

Multiply $2 x$ by $1$ .

Multiply $3$ by $1$ .

Simplify by adding terms.

Add $12 x^{8}$ and $15 x^{8}$ .

$18 x^{7} - 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 4 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 27 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x - 4 x^{6} - 6 x^{5} + 2 x + 3$

Add $18 x^{7}$ and $4 x^{7}$ .

$- 2 x^{12} - 3 x^{11} - 2 x^{10} - 3 x^{9} + 22 x^{7} + 6 x^{6} - 6 x^{11} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 27 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x - 4 x^{6} - 6 x^{5} + 2 x + 3$

Subtract $6 x^{11}$ from $- 3 x^{11}$ .

$- 2 x^{12} - 9 x^{11} - 2 x^{10} - 3 x^{9} + 22 x^{7} + 6 x^{6} - 9 x^{10} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 27 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x - 4 x^{6} - 6 x^{5} + 2 x + 3$

Subtract $9 x^{10}$ from $- 2 x^{10}$ .

$- 2 x^{12} - 9 x^{11} - 11 x^{10} - 3 x^{9} + 22 x^{7} + 6 x^{6} - 12 x^{5} - 18 x^{4} + 10 x^{9} + 27 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x - 4 x^{6} - 6 x^{5} + 2 x + 3$

Add $- 3 x^{9}$ and $10 x^{9}$ .

$- 2 x^{12} - 9 x^{11} - 11 x^{10} + 7 x^{9} + 22 x^{7} + 6 x^{6} - 12 x^{5} - 18 x^{4} + 27 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x - 4 x^{6} - 6 x^{5} + 2 x + 3$

Subtract $4 x^{6}$ from $6 x^{6}$ .

$- 2 x^{12} - 9 x^{11} - 11 x^{10} + 7 x^{9} + 22 x^{7} + 2 x^{6} - 12 x^{5} - 18 x^{4} + 27 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x - 6 x^{5} + 2 x + 3$

Subtract $6 x^{5}$ from $- 12 x^{5}$ .

$- 2 x^{12} - 9 x^{11} - 11 x^{10} + 7 x^{9} + 22 x^{7} + 2 x^{6} - 18 x^{5} - 18 x^{4} + 27 x^{8} + 2 x^{3} + 3 x^{2} - 10 x^{4} - 15 x^{3} + 6 x^{2} + 9 x + 2 x + 3$

Subtract $10 x^{4}$ from $- 18 x^{4}$ .

$- 2 x^{12} - 9 x^{11} - 11 x^{10} + 7 x^{9} + 22 x^{7} + 2 x^{6} - 18 x^{5} - 28 x^{4} + 27 x^{8} + 2 x^{3} + 3 x^{2} - 15 x^{3} + 6 x^{2} + 9 x + 2 x + 3$

Subtract $15 x^{3}$ from $2 x^{3}$ .

$- 2 x^{12} - 9 x^{11} - 11 x^{10} + 7 x^{9} + 22 x^{7} + 2 x^{6} - 18 x^{5} - 28 x^{4} + 27 x^{8} - 13 x^{3} + 3 x^{2} + 6 x^{2} + 9 x + 2 x + 3$

Add $3 x^{2}$ and $6 x^{2}$ .

$- 2 x^{12} - 9 x^{11} - 11 x^{10} + 7 x^{9} + 22 x^{7} + 2 x^{6} - 18 x^{5} - 28 x^{4} + 27 x^{8} - 13 x^{3} + 9 x^{2} + 9 x + 2 x + 3$

Add $9 x$ and $2 x$ .

$- 2 x^{12} - 9 x^{11} - 11 x^{10} + 7 x^{9} + 22 x^{7} + 2 x^{6} - 18 x^{5} - 28 x^{4} + 27 x^{8} - 13 x^{3} + 9 x^{2} + 11 x + 3$

Reorder.

Move $- 28 x^{4}$ .

$- 2 x^{12} - 9 x^{11} - 11 x^{10} + 7 x^{9} + 22 x^{7} + 2 x^{6} - 18 x^{5} + 27 x^{8} - 28 x^{4} - 13 x^{3} + 9 x^{2} + 11 x + 3$

Move $- 18 x^{5}$ .

$- 2 x^{12} - 9 x^{11} - 11 x^{10} + 7 x^{9} + 22 x^{7} + 2 x^{6} + 27 x^{8} - 18 x^{5} - 28 x^{4} - 13 x^{3} + 9 x^{2} + 11 x + 3$

Move $2 x^{6}$ .

$- 2 x^{12} - 9 x^{11} - 11 x^{10} + 7 x^{9} + 22 x^{7} + 27 x^{8} + 2 x^{6} - 18 x^{5} - 28 x^{4} - 13 x^{3} + 9 x^{2} + 11 x + 3$

Move $22 x^{7}$ .

$- 2 x^{12} - 9 x^{11} - 11 x^{10} + 7 x^{9} + 27 x^{8} + 22 x^{7} + 2 x^{6} - 18 x^{5} - 28 x^{4} - 13 x^{3} + 9 x^{2} + 11 x + 3$

Step 2

The degree of a polynomial is the highest degree of its terms.

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$- 2 x^{12} \to 12$

$- 9 x^{11} \to 11$

$- 11 x^{10} \to 10$

$7 x^{9} \to 9$

$27 x^{8} \to 8$

$22 x^{7} \to 7$

$2 x^{6} \to 6$

$- 18 x^{5} \to 5$

$- 28 x^{4} \to 4$

$- 13 x^{3} \to 3$

$9 x^{2} \to 2$

$11 x \to 1$

$3 \to 0$

The largest exponent is the degree of the polynomial.

$12$

Step 3

The leading term in a polynomial is the term with the highest degree.

$- 2 x^{12}$

Step 4

The leading coefficient of a polynomial is the coefficient of the leading term.

The leading term in a polynomial is the term with the highest degree.

$- 2 x^{12}$

The leading coefficient in a polynomial is the coefficient of the leading term.

$- 2$

Step 5

List the results.

Polynomial Degree: $12$

Leading Term: $- 2 x^{12}$

Leading Coefficient: $- 2$

Find the Degree, Leading Term, and Leading Coefficient (x+1)^3(x^2-1)^2(1-x^4)(2x+3)

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