Algebra Examples

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

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c$

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Move $x^{2}$ .

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

Step 3.1.1.2

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

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

Step 3.1.1.3

Add $2$ and $2$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

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

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Step 3.1.3.1

Move $x$ .

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

Step 3.1.3.2

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

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Step 3.1.3.2.1

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

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

Step 3.1.3.2.2

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

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

Step 3.1.3.3

Add $1$ and $2$ .

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

Step 3.1.4

Multiply $2$ by $4$ .

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

Step 3.1.5

Multiply $- 4$ by $2$ .

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

Step 3.1.6

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

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Step 3.1.6.1

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

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Step 3.1.6.1.1

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

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

Step 3.1.6.1.2

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

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

Step 3.1.6.2

Add $1$ and $2$ .

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

Step 3.1.7

Rewrite using the commutative property of multiplication.

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

Step 3.1.8

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

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Step 3.1.8.1

Move $x$ .

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

Step 3.1.8.2

Multiply $x$ by $x$ .

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

Step 3.1.9

Move $- 4$ to the left of $x$ .

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

Step 3.1.10

Multiply $4$ by $4$ .

$2 x^{4} + 8 x^{3} - 8 x^{2} + x^{3} + 4 x^{2} - 4 x + 4 x^{2} + 16 x + 4 \cdot - 4$

Step 3.1.11

Multiply $4$ by $- 4$ .

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

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Add $8 x^{3}$ and $x^{3}$ .

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

Step 3.2.2

Add $- 8 x^{2}$ and $4 x^{2}$ .

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

Step 3.2.3

Combine the opposite terms in $2 x^{4} + 9 x^{3} - 4 x^{2} - 4 x + 4 x^{2} + 16 x - 16$ .

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Step 3.2.3.1

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

$2 x^{4} + 9 x^{3} - 4 x + 0 + 16 x - 16$

Step 3.2.3.2

Add $2 x^{4} + 9 x^{3} - 4 x$ and $0$ .

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

Step 3.2.4

Add $- 4 x$ and $16 x$ .

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