Simplify (3x^2-4x+1)(2x-3x^2+5)

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

Step 1

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

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

Step 2

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $3$ by $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Multiply $3$ by $- 3$ .

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

Multiply $5$ by $3$ .

$6 x^{3} - 9 x^{4} + 15 x^{2} - 4 x (2 x) - 4 x (- 3 x^{2}) - 4 x \cdot 5 + 1 (2 x) + 1 (- 3 x^{2}) + 1 \cdot 5$

Rewrite using the commutative property of multiplication.

$6 x^{3} - 9 x^{4} + 15 x^{2} - 4 \cdot 2 x \cdot x - 4 x (- 3 x^{2}) - 4 x \cdot 5 + 1 (2 x) + 1 (- 3 x^{2}) + 1 \cdot 5$

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

Move $x$ .

$6 x^{3} - 9 x^{4} + 15 x^{2} - 4 \cdot 2 (x \cdot x) - 4 x (- 3 x^{2}) - 4 x \cdot 5 + 1 (2 x) + 1 (- 3 x^{2}) + 1 \cdot 5$

Multiply $x$ by $x$ .

$6 x^{3} - 9 x^{4} + 15 x^{2} - 4 \cdot 2 x^{2} - 4 x (- 3 x^{2}) - 4 x \cdot 5 + 1 (2 x) + 1 (- 3 x^{2}) + 1 \cdot 5$

Multiply $- 4$ by $2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $- 4$ by $- 3$ .

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

Multiply $5$ by $- 4$ .

$6 x^{3} - 9 x^{4} + 15 x^{2} - 8 x^{2} + 12 x^{3} - 20 x + 1 (2 x) + 1 (- 3 x^{2}) + 1 \cdot 5$

Multiply $2 x$ by $1$ .

$6 x^{3} - 9 x^{4} + 15 x^{2} - 8 x^{2} + 12 x^{3} - 20 x + 2 x + 1 (- 3 x^{2}) + 1 \cdot 5$

Multiply $- 3 x^{2}$ by $1$ .

$6 x^{3} - 9 x^{4} + 15 x^{2} - 8 x^{2} + 12 x^{3} - 20 x + 2 x - 3 x^{2} + 1 \cdot 5$

Multiply $5$ by $1$ .

$6 x^{3} - 9 x^{4} + 15 x^{2} - 8 x^{2} + 12 x^{3} - 20 x + 2 x - 3 x^{2} + 5$

Simplify by adding terms.

Add $6 x^{3}$ and $12 x^{3}$ .

$18 x^{3} - 9 x^{4} + 15 x^{2} - 8 x^{2} - 20 x + 2 x - 3 x^{2} + 5$

Subtract $8 x^{2}$ from $15 x^{2}$ .

$18 x^{3} - 9 x^{4} + 7 x^{2} - 20 x + 2 x - 3 x^{2} + 5$

Subtract $3 x^{2}$ from $7 x^{2}$ .

$18 x^{3} - 9 x^{4} + 4 x^{2} - 20 x + 2 x + 5$

Add $- 20 x$ and $2 x$ .

$18 x^{3} - 9 x^{4} + 4 x^{2} - 18 x + 5$

Reorder $18 x^{3}$ and $- 9 x^{4}$ .

$- 9 x^{4} + 18 x^{3} + 4 x^{2} - 18 x + 5$

Simplify (3x^2-4x+1)(2x-3x^2+5)

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