Pre-Algebra Examples

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

Step 1

Factor using the perfect square rule.

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Step 1.1

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

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

Step 1.2

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 1.3

Rewrite the polynomial.

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

Step 1.4

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

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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 $b = 2$ .

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x + 2$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Multiply $\frac{{(x - 2)}^{2}}{x - 2}$ by $\frac{1}{x^{2} + 2 x + 4}$ .

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

Step 3.3

Cancel the common factor of ${(x - 2)}^{2}$ and $x - 2$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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