Simplify ((x^2+5x+6)/(x-6))÷((x^2-9)/(2x-12))

$\frac{x^{2} + 5 x + 6}{x - 6} \div \frac{x^{2} - 9}{2 x - 12}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 5 x + 6}{x - 6} \cdot \frac{2 x - 12}{x^{2} - 9}$

Step 2

Factor $x^{2} + 5 x + 6$ 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 $6$ and whose sum is $5$ .

$2, 3$

Write the factored form using these integers.

$\frac{(x + 2) (x + 3)}{x - 6} \cdot \frac{2 x - 12}{x^{2} - 9}$

Step 3

Factor $2$ out of $2 x - 12$ .

Factor $2$ out of $2 x$ .

$\frac{(x + 2) (x + 3)}{x - 6} \cdot \frac{2 (x) - 12}{x^{2} - 9}$

Factor $2$ out of $- 12$ .

$\frac{(x + 2) (x + 3)}{x - 6} \cdot \frac{2 x + 2 \cdot - 6}{x^{2} - 9}$

Factor $2$ out of $2 x + 2 \cdot - 6$ .

$\frac{(x + 2) (x + 3)}{x - 6} \cdot \frac{2 (x - 6)}{x^{2} - 9}$

Step 4

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

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

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 $b = 3$ .

$\frac{(x + 2) (x + 3)}{x - 6} \cdot \frac{2 (x - 6)}{(x + 3) (x - 3)}$

Step 5

Simplify terms.

Cancel the common factor of $x + 3$ .

Factor $x + 3$ out of $(x + 2) (x + 3)$ .

$\frac{(x + 3) (x + 2)}{x - 6} \cdot \frac{2 (x - 6)}{(x + 3) (x - 3)}$

Cancel the common factor.

$\frac{(x + 3) (x + 2)}{x - 6} \cdot \frac{2 (x - 6)}{(x + 3) (x - 3)}$

Rewrite the expression.

$\frac{x + 2}{x - 6} \cdot \frac{2 (x - 6)}{x - 3}$

Cancel the common factor of $x - 6$ .

Factor $x - 6$ out of $2 (x - 6)$ .

$\frac{x + 2}{x - 6} \cdot \frac{(x - 6) \cdot 2}{x - 3}$

Cancel the common factor.

$\frac{x + 2}{x - 6} \cdot \frac{(x - 6) \cdot 2}{x - 3}$

Rewrite the expression.

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

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

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

Move $2$ to the left of $x + 2$ .

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

Simplify ((x^2+5x+6)/(x-6))÷((x^2-9)/(2x-12))

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