Find the End Behavior f(x)=(x-2)^2(x+3)(x+1)^2

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

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

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

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

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 2$ to the left of $x$ .

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

Multiply $- 2$ by $- 2$ .

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

Subtract $2 x$ from $- 2 x$ .

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

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

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

Simplify terms.

Simplify each term.

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $- 4$ .

$(x^{3} + 3 x^{2} - 4 x^{2} - 12 x + 4 x + 4 \cdot 3) {(x + 1)}^{2}$

Multiply $4$ by $3$ .

$(x^{3} + 3 x^{2} - 4 x^{2} - 12 x + 4 x + 12) {(x + 1)}^{2}$

Simplify by adding terms.

Subtract $4 x^{2}$ from $3 x^{2}$ .

$(x^{3} - x^{2} - 12 x + 4 x + 12) {(x + 1)}^{2}$

Add $- 12 x$ and $4 x$ .

$(x^{3} - x^{2} - 8 x + 12) {(x + 1)}^{2}$

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

$(x^{3} - x^{2} - 8 x + 12) ((x + 1) (x + 1))$

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$(x^{3} - x^{2} - 8 x + 12) (x (x + 1) + 1 (x + 1))$

Apply the distributive property.

$(x^{3} - x^{2} - 8 x + 12) (x \cdot x + x \cdot 1 + 1 (x + 1))$

Apply the distributive property.

$(x^{3} - x^{2} - 8 x + 12) (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(x^{3} - x^{2} - 8 x + 12) (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1)$

Multiply $x$ by $1$ .

$(x^{3} - x^{2} - 8 x + 12) (x^{2} + x + 1 x + 1 \cdot 1)$

Multiply $x$ by $1$ .

$(x^{3} - x^{2} - 8 x + 12) (x^{2} + x + x + 1 \cdot 1)$

Multiply $1$ by $1$ .

$(x^{3} - x^{2} - 8 x + 12) (x^{2} + x + x + 1)$

Add $x$ and $x$ .

$(x^{3} - x^{2} - 8 x + 12) (x^{2} + 2 x + 1)$

Expand $(x^{3} - x^{2} - 8 x + 12) (x^{2} + 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} x^{2} + x^{3} (2 x) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Simplify terms.

Simplify each term.

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

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

$x^{3 + 2} + x^{3} (2 x) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Add $3$ and $2$ .

$x^{5} + x^{3} (2 x) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{5} + 2 x^{3} x + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

Move $x$ .

$x^{5} + 2 (x \cdot x^{3}) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

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

$x^{5} + 2 (x^{1} x^{3}) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

$x^{5} + 2 x^{1 + 3} + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Add $1$ and $3$ .

$x^{5} + 2 x^{4} + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

$x^{5} + 2 x^{4} + x^{3} - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

Move $x^{2}$ .

$x^{5} + 2 x^{4} + x^{3} - (x^{2} x^{2}) - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

$x^{5} + 2 x^{4} + x^{3} - x^{2 + 2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Add $2$ and $2$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot 2 x^{2} x - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

Move $x$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

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

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Add $1$ and $2$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot 2 x^{3} - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Multiply $- 1$ by $2$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Multiply $- 1$ by $1$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

Move $x^{2}$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 (x^{2} x) - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

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

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 (x^{2} x^{1}) - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{2 + 1} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Add $2$ and $1$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 8 \cdot 2 x \cdot x - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

Move $x$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 8 \cdot 2 (x \cdot x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Multiply $x$ by $x$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 8 \cdot 2 x^{2} - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Multiply $- 8$ by $2$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Multiply $- 8$ by $1$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Multiply $2$ by $12$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12 \cdot 1$

Multiply $12$ by $1$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Simplify by adding terms.

Subtract $x^{4}$ from $2 x^{4}$ .

$x^{5} + x^{4} + x^{3} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

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

$x^{5} + x^{4} - x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Subtract $8 x^{3}$ from $- x^{3}$ .

$x^{5} + x^{4} - 9 x^{3} - x^{2} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Subtract $16 x^{2}$ from $- x^{2}$ .

$x^{5} + x^{4} - 9 x^{3} - 17 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Add $- 17 x^{2}$ and $12 x^{2}$ .

$x^{5} + x^{4} - 9 x^{3} - 5 x^{2} - 8 x + 24 x + 12$

Add $- 8 x$ and $24 x$ .

$x^{5} + x^{4} - 9 x^{3} - 5 x^{2} + 16 x + 12$

The largest exponent is the degree of the polynomial.

$5$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

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

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

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

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 2$ to the left of $x$ .

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

Multiply $- 2$ by $- 2$ .

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

Subtract $2 x$ from $- 2 x$ .

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

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

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

Simplify terms.

Simplify each term.

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $- 4$ .

$(x^{3} + 3 x^{2} - 4 x^{2} - 12 x + 4 x + 4 \cdot 3) {(x + 1)}^{2}$

Multiply $4$ by $3$ .

$(x^{3} + 3 x^{2} - 4 x^{2} - 12 x + 4 x + 12) {(x + 1)}^{2}$

Simplify by adding terms.

Subtract $4 x^{2}$ from $3 x^{2}$ .

$(x^{3} - x^{2} - 12 x + 4 x + 12) {(x + 1)}^{2}$

Add $- 12 x$ and $4 x$ .

$(x^{3} - x^{2} - 8 x + 12) {(x + 1)}^{2}$

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

$(x^{3} - x^{2} - 8 x + 12) ((x + 1) (x + 1))$

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$(x^{3} - x^{2} - 8 x + 12) (x (x + 1) + 1 (x + 1))$

Apply the distributive property.

$(x^{3} - x^{2} - 8 x + 12) (x \cdot x + x \cdot 1 + 1 (x + 1))$

Apply the distributive property.

$(x^{3} - x^{2} - 8 x + 12) (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(x^{3} - x^{2} - 8 x + 12) (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1)$

Multiply $x$ by $1$ .

$(x^{3} - x^{2} - 8 x + 12) (x^{2} + x + 1 x + 1 \cdot 1)$

Multiply $x$ by $1$ .

$(x^{3} - x^{2} - 8 x + 12) (x^{2} + x + x + 1 \cdot 1)$

Multiply $1$ by $1$ .

$(x^{3} - x^{2} - 8 x + 12) (x^{2} + x + x + 1)$

Add $x$ and $x$ .

$(x^{3} - x^{2} - 8 x + 12) (x^{2} + 2 x + 1)$

Expand $(x^{3} - x^{2} - 8 x + 12) (x^{2} + 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} x^{2} + x^{3} (2 x) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Simplify terms.

Simplify each term.

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

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

$x^{3 + 2} + x^{3} (2 x) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Add $3$ and $2$ .

$x^{5} + x^{3} (2 x) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{5} + 2 x^{3} x + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

Move $x$ .

$x^{5} + 2 (x \cdot x^{3}) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

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

$x^{5} + 2 (x^{1} x^{3}) + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

$x^{5} + 2 x^{1 + 3} + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Add $1$ and $3$ .

$x^{5} + 2 x^{4} + x^{3} \cdot 1 - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

$x^{5} + 2 x^{4} + x^{3} - x^{2} x^{2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

Move $x^{2}$ .

$x^{5} + 2 x^{4} + x^{3} - (x^{2} x^{2}) - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

$x^{5} + 2 x^{4} + x^{3} - x^{2 + 2} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Add $2$ and $2$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - x^{2} (2 x) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot 2 x^{2} x - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

Move $x$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

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

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Add $1$ and $2$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 1 \cdot 2 x^{3} - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Multiply $- 1$ by $2$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} \cdot 1 - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Multiply $- 1$ by $1$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x \cdot x^{2} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

Move $x^{2}$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 (x^{2} x) - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

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

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 (x^{2} x^{1}) - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{2 + 1} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Add $2$ and $1$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 8 x (2 x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Rewrite using the commutative property of multiplication.

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 8 \cdot 2 x \cdot x - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

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

Move $x$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 8 \cdot 2 (x \cdot x) - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Multiply $x$ by $x$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 8 \cdot 2 x^{2} - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Multiply $- 8$ by $2$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x \cdot 1 + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Multiply $- 8$ by $1$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 12 (2 x) + 12 \cdot 1$

Multiply $2$ by $12$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12 \cdot 1$

Multiply $12$ by $1$ .

$x^{5} + 2 x^{4} + x^{3} - x^{4} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Simplify by adding terms.

Subtract $x^{4}$ from $2 x^{4}$ .

$x^{5} + x^{4} + x^{3} - 2 x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

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

$x^{5} + x^{4} - x^{3} - x^{2} - 8 x^{3} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Subtract $8 x^{3}$ from $- x^{3}$ .

$x^{5} + x^{4} - 9 x^{3} - x^{2} - 16 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Subtract $16 x^{2}$ from $- x^{2}$ .

$x^{5} + x^{4} - 9 x^{3} - 17 x^{2} - 8 x + 12 x^{2} + 24 x + 12$

Add $- 17 x^{2}$ and $12 x^{2}$ .

$x^{5} + x^{4} - 9 x^{3} - 5 x^{2} - 8 x + 24 x + 12$

Add $- 8 x$ and $24 x$ .

$x^{5} + x^{4} - 9 x^{3} - 5 x^{2} + 16 x + 12$

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

$x^{5}$

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

$1$

Step 4

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Falls to the left and rises to the right

Step 7

Find the End Behavior f(x)=(x-2)^2(x+3)(x+1)^2

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