Find the Product (3x^2+x-2)(-4x^2-2x-1)

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

Step 1

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

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

Step 2

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Multiply $3$ by $- 4$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $3$ by $- 2$ .

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

Multiply $- 1$ by $3$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 4$ by $- 2$ .

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

Multiply $- 2$ by $- 2$ .

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

Multiply $- 2$ by $- 1$ .

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

Simplify by adding terms.

Subtract $4 x^{3}$ from $- 6 x^{3}$ .

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

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

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

Add $- 5 x^{2}$ and $8 x^{2}$ .

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

Add $- x$ and $4 x$ .

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

Find the Product (3x^2+x-2)(-4x^2-2x-1)

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