Algebra Examples

$\frac{4 - x^{2}}{- x - 3} \cdot \frac{x + 4}{x - 2} \cdot \frac{- x - 3}{x^{2} + 6 x + 8}$

Step 1

Factor $x^{2} + 6 x + 8$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $8$ and whose sum is $6$ .

$2, 4$

Step 1.2

Write the factored form using these integers.

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

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 $a = 2$ and $b = x$ .

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{(2 + x) (2 - x)}{- x - 3}$ by $\frac{x + 4}{x - 2}$ .

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

Step 3.2

Cancel the common factor of $x + 4$ .

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Step 3.2.1

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

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

Step 3.2.2

Factor $x + 4$ out of $(x + 2) (x + 4)$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $- x - 3$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

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

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

Step 3.5

Cancel the common factor of $2 + x$ and $x + 2$ .

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Step 3.5.1

Reorder terms.

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

Step 3.5.2

Cancel the common factor.

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

Step 3.5.3

Rewrite the expression.

$\frac{2 - x}{x - 2}$

Step 3.6

Cancel the common factor of $2 - x$ and $x - 2$ .

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Step 3.6.1

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- 1 (- 2) - x}{x - 2}$

Step 3.6.2

Factor $- 1$ out of $- x$ .

$\frac{- 1 (- 2) - (x)}{x - 2}$

Step 3.6.3

Factor $- 1$ out of $- 1 (- 2) - (x)$ .

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

Step 3.6.4

Reorder terms.

$\frac{- 1 (x - 2)}{x - 2}$

Step 3.6.5

Cancel the common factor.

$\frac{- 1 (x - 2)}{x - 2}$

Step 3.6.6

Divide $- 1$ by $1$ .

$- 1$