Simplify ((x+2)/(x^2+5x+6))÷((3x+1)/(x^2-9))

$\frac{x + 2}{x^{2} + 5 x + 6} \div \frac{3 x + 1}{x^{2} - 9}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x + 2}{x^{2} + 5 x + 6} \cdot \frac{x^{2} - 9}{3 x + 1}$

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.

$\frac{x + 2}{(x + 2) (x + 3)} \cdot \frac{x^{2} - 9}{3 x + 1}$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{x + 2}{(x + 2) (x + 3)} \cdot \frac{x^{2} - 9}{3 x + 1}$

Rewrite the expression.

$\frac{1}{x + 3} \cdot \frac{x^{2} - 9}{3 x + 1}$

Simplify the numerator.

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

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

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 $b = 3$ .

$\frac{1}{x + 3} \cdot \frac{(x + 3) (x - 3)}{3 x + 1}$

Cancel the common factor of $x + 3$ .

Cancel the common factor.

$\frac{1}{x + 3} \cdot \frac{(x + 3) (x - 3)}{3 x + 1}$

Rewrite the expression.

$\frac{x - 3}{3 x + 1}$

Simplify ((x+2)/(x^2+5x+6))÷((3x+1)/(x^2-9))

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