Find the Product (3-2x+5x^2)(4-x+2x^2)

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

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

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

Simplify terms.

Simplify each term.

Multiply $3$ by $4$ .

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

Multiply $- 1$ by $3$ .

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

Multiply $2$ by $3$ .

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

Multiply $4$ by $- 2$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $- 2$ by $2$ .

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

Multiply $4$ by $5$ .

$12 - 3 x + 6 x^{2} - 8 x + 2 x^{2} - 4 x^{3} + 20 x^{2} + 5 x^{2} (- x) + 5 x^{2} (2 x^{2})$

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $5$ by $- 1$ .

$12 - 3 x + 6 x^{2} - 8 x + 2 x^{2} - 4 x^{3} + 20 x^{2} - 5 x^{3} + 5 x^{2} (2 x^{2})$

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Multiply $5$ by $2$ .

$12 - 3 x + 6 x^{2} - 8 x + 2 x^{2} - 4 x^{3} + 20 x^{2} - 5 x^{3} + 10 x^{4}$

Simplify by adding terms.

Subtract $8 x$ from $- 3 x$ .

$12 + 6 x^{2} - 11 x + 2 x^{2} - 4 x^{3} + 20 x^{2} - 5 x^{3} + 10 x^{4}$

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

$12 + 8 x^{2} - 11 x - 4 x^{3} + 20 x^{2} - 5 x^{3} + 10 x^{4}$

Add $8 x^{2}$ and $20 x^{2}$ .

$12 + 28 x^{2} - 11 x - 4 x^{3} - 5 x^{3} + 10 x^{4}$

Subtract $5 x^{3}$ from $- 4 x^{3}$ .

$12 + 28 x^{2} - 11 x - 9 x^{3} + 10 x^{4}$

Find the Product (3-2x+5x^2)(4-x+2x^2)

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