Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Factor $2 (x + 1)$ out of $2 x {(x + 1)}^{2} - 4 (x + 1)$ .

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Step 1.1.1

Factor $2 (x + 1)$ out of $2 x {(x + 1)}^{2}$ .

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

Step 1.1.2

Factor $2 (x + 1)$ out of $- 4 (x + 1)$ .

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

Step 1.1.3

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

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Multiply $x$ by $x$ .

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

Step 1.4

Multiply $x$ by $1$ .

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

Step 1.5

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

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Step 1.5.1

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 1.5.2

Write the factored form using these integers.

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

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

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

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 $b = 1$ .

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $2$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x + 1$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x - 1$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Divide $x + 2$ by $1$ .

$x + 2$