Precalculus Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

Move $x^{2}$ .

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

Step 2.1.1.2

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

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

Step 2.1.1.3

Add $2$ and $2$ .

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

Step 2.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.1.3

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

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Step 2.1.3.1

Move $x$ .

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

Step 2.1.3.2

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

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Step 2.1.3.2.1

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

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

Step 2.1.3.2.2

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

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

Step 2.1.3.3

Add $1$ and $2$ .

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

Step 2.1.4

Multiply $2$ by $3$ .

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

Step 2.1.5

Multiply $2$ by $2$ .

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

Step 2.1.6

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

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Step 2.1.6.1

Move $x^{2}$ .

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

Step 2.1.6.2

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

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Step 2.1.6.2.1

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

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

Step 2.1.6.2.2

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

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

Step 2.1.6.3

Add $2$ and $1$ .

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

Step 2.1.7

Rewrite using the commutative property of multiplication.

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

Step 2.1.8

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

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Step 2.1.8.1

Move $x$ .

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

Step 2.1.8.2

Multiply $x$ by $x$ .

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

Step 2.1.9

Multiply $- 1$ by $3$ .

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

Step 2.1.10

Multiply $2$ by $- 1$ .

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

Step 2.1.11

Multiply $3$ by $4$ .

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

Step 2.1.12

Multiply $4$ by $2$ .

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Add $x^{2}$ and $4 x^{2}$ .

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

Step 2.2.4

Add $- 2 x$ and $12 x$ .

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