Algebra Examples

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

Step 1

Factor $x^{2} - 4$ out of $x^{2} (x^{2} - 4) + 4 x (x^{2} - 4) + 4 (x^{2} - 4)$ .

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Step 1.1

Factor $x^{2} - 4$ out of $x^{2} (x^{2} - 4)$ .

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

Step 1.2

Factor $x^{2} - 4$ out of $4 x (x^{2} - 4)$ .

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

Step 1.3

Factor $x^{2} - 4$ out of $4 (x^{2} - 4)$ .

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

Step 1.4

Factor $x^{2} - 4$ out of $(x^{2} - 4) x^{2} + (x^{2} - 4) (4 x)$ .

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

Step 1.5

Factor $x^{2} - 4$ out of $(x^{2} - 4) (x^{2} + 4 x) + (x^{2} - 4) \cdot 4$ .

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

Step 2

Rewrite $4$ as $2^{2}$ .

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

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 4

Factor using the perfect square rule.

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Step 4.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 4.3

Rewrite the polynomial.

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

Step 4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 5

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

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Step 5.1

Move ${(x + 2)}^{2}$ .

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

Step 5.2

Multiply ${(x + 2)}^{2}$ by $x + 2$ .

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Step 5.2.1

Raise $x + 2$ to the power of $1$ .

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

Step 5.2.2

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

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

Step 5.3

Add $2$ and $1$ .

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