Algebra Examples

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

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

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

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $4 x^{2} x^{2} + 4 x^{2} x + 4 x^{2} \cdot 2 + x \cdot x^{2} + x \cdot x + x \cdot 2 - 2 x^{2} - 2 x - 2 \cdot 2$ .

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Step 3.1.1

Reorder the factors in the terms $x \cdot 2$ and $- 2 x$ .

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

Step 3.1.2

Subtract $2 x$ from $2 x$ .

$4 x^{2} x^{2} + 4 x^{2} x + 4 x^{2} \cdot 2 + x \cdot x^{2} + x \cdot x - 2 x^{2} + 0 - 2 \cdot 2$

Step 3.1.3

Add $4 x^{2} x^{2} + 4 x^{2} x + 4 x^{2} \cdot 2 + x \cdot x^{2} + x \cdot x - 2 x^{2}$ and $0$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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Step 3.2.1.1

Move $x^{2}$ .

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Add $2$ and $2$ .

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

Step 3.2.2

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

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Step 3.2.2.1

Move $x$ .

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

Step 3.2.2.2

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

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Step 3.2.2.2.1

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

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

Step 3.2.2.2.2

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

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

Step 3.2.2.3

Add $1$ and $2$ .

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

Step 3.2.3

Multiply $2$ by $4$ .

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

Step 3.2.4

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

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Step 3.2.4.1

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

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Step 3.2.4.1.1

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

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

Step 3.2.4.1.2

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

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

Step 3.2.4.2

Add $1$ and $2$ .

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

Step 3.2.5

Multiply $x$ by $x$ .

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

Step 3.2.6

Multiply $- 2$ by $2$ .

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

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Add $4 x^{3}$ and $x^{3}$ .

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

Step 3.3.2

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

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

Step 3.3.3

Subtract $2 x^{2}$ from $9 x^{2}$ .

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