Simplify ((x^2+3x+2)/(x^2-1))÷((x^2+6x)/(x^2-x))

$\frac{x^{2} + 3 x + 2}{x^{2} - 1} \div \frac{x^{2} + 6 x}{x^{2} - x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 3 x + 2}{x^{2} - 1} \cdot \frac{x^{2} - x}{x^{2} + 6 x}$

Step 2

Factor $x^{2} + 3 x + 2$ using the AC method.

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 $3$ .

$1, 2$

Write the factored form using these integers.

$\frac{(x + 1) (x + 2)}{x^{2} - 1} \cdot \frac{x^{2} - x}{x^{2} + 6 x}$

Step 3

Simplify the denominator.

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

$\frac{(x + 1) (x + 2)}{x^{2} - 1^{2}} \cdot \frac{x^{2} - x}{x^{2} + 6 x}$

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 $b = 1$ .

$\frac{(x + 1) (x + 2)}{(x + 1) (x - 1)} \cdot \frac{x^{2} - x}{x^{2} + 6 x}$

Step 4

Simplify terms.

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$\frac{(x + 1) (x + 2)}{(x + 1) (x - 1)} \cdot \frac{x^{2} - x}{x^{2} + 6 x}$

Rewrite the expression.

$\frac{x + 2}{x - 1} \cdot \frac{x^{2} - x}{x^{2} + 6 x}$

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

Factor $x$ out of $x^{2}$ .

$\frac{x + 2}{x - 1} \cdot \frac{x \cdot x - x}{x^{2} + 6 x}$

Factor $x$ out of $- x$ .

$\frac{x + 2}{x - 1} \cdot \frac{x \cdot x + x \cdot - 1}{x^{2} + 6 x}$

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$\frac{x + 2}{x - 1} \cdot \frac{x (x - 1)}{x^{2} + 6 x}$

Factor $x$ out of $x^{2} + 6 x$ .

Factor $x$ out of $x^{2}$ .

$\frac{x + 2}{x - 1} \cdot \frac{x (x - 1)}{x \cdot x + 6 x}$

Factor $x$ out of $6 x$ .

$\frac{x + 2}{x - 1} \cdot \frac{x (x - 1)}{x \cdot x + x \cdot 6}$

Factor $x$ out of $x \cdot x + x \cdot 6$ .

$\frac{x + 2}{x - 1} \cdot \frac{x (x - 1)}{x (x + 6)}$

Cancel the common factor of $x - 1$ .

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

$\frac{x + 2}{x - 1} \cdot \frac{(x - 1) x}{x (x + 6)}$

Cancel the common factor.

$\frac{x + 2}{x - 1} \cdot \frac{(x - 1) x}{x (x + 6)}$

Rewrite the expression.

$(x + 2) \frac{x}{x (x + 6)}$

Cancel the common factor of $x$ .

Cancel the common factor.

$(x + 2) \frac{x}{x (x + 6)}$

Rewrite the expression.

$(x + 2) \frac{1}{x + 6}$

Multiply $x + 2$ by $\frac{1}{x + 6}$ .

$\frac{x + 2}{x + 6}$

Simplify ((x^2+3x+2)/(x^2-1))÷((x^2+6x)/(x^2-x))

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