Simplify ((9-a^2)/(a^2+5a+6))÷((2a-6)/(5a+10))

$\frac{9 - a^{2}}{a^{2} + 5 a + 6} \div \frac{2 a - 6}{5 a + 10}$

To divide by a fraction, multiply by its reciprocal.

$\frac{9 - a^{2}}{a^{2} + 5 a + 6} \cdot \frac{5 a + 10}{2 a - 6}$

Simplify the numerator.

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

$\frac{3^{2} - a^{2}}{a^{2} + 5 a + 6} \cdot \frac{5 a + 10}{2 a - 6}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = a$ .

$\frac{(3 + a) (3 - a)}{a^{2} + 5 a + 6} \cdot \frac{5 a + 10}{2 a - 6}$

Factor $a^{2} + 5 a + 6$ 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.

$\frac{(3 + a) (3 - a)}{(a + 2) (a + 3)} \cdot \frac{5 a + 10}{2 a - 6}$

Simplify terms.

Cancel the common factor of $3 + a$ and $a + 3$ .

Reorder terms.

$\frac{(a + 3) (3 - a)}{(a + 2) (a + 3)} \cdot \frac{5 a + 10}{2 a - 6}$

Cancel the common factor.

$\frac{(a + 3) (3 - a)}{(a + 2) (a + 3)} \cdot \frac{5 a + 10}{2 a - 6}$

Rewrite the expression.

$\frac{3 - a}{a + 2} \cdot \frac{5 a + 10}{2 a - 6}$

Factor $5$ out of $5 a + 10$ .

Factor $5$ out of $5 a$ .

$\frac{3 - a}{a + 2} \cdot \frac{5 (a) + 10}{2 a - 6}$

Factor $5$ out of $10$ .

$\frac{3 - a}{a + 2} \cdot \frac{5 a + 5 \cdot 2}{2 a - 6}$

Factor $5$ out of $5 a + 5 \cdot 2$ .

$\frac{3 - a}{a + 2} \cdot \frac{5 (a + 2)}{2 a - 6}$

Factor $2$ out of $2 a - 6$ .

Factor $2$ out of $2 a$ .

$\frac{3 - a}{a + 2} \cdot \frac{5 (a + 2)}{2 (a) - 6}$

Factor $2$ out of $- 6$ .

$\frac{3 - a}{a + 2} \cdot \frac{5 (a + 2)}{2 a + 2 \cdot - 3}$

Factor $2$ out of $2 a + 2 \cdot - 3$ .

$\frac{3 - a}{a + 2} \cdot \frac{5 (a + 2)}{2 (a - 3)}$

Cancel the common factor of $a + 2$ .

Factor $a + 2$ out of $5 (a + 2)$ .

$\frac{3 - a}{a + 2} \cdot \frac{(a + 2) \cdot 5}{2 (a - 3)}$

Cancel the common factor.

$\frac{3 - a}{a + 2} \cdot \frac{(a + 2) \cdot 5}{2 (a - 3)}$

Rewrite the expression.

$(3 - a) \frac{5}{2 (a - 3)}$

Multiply $3 - a$ by $\frac{5}{2 (a - 3)}$ .

$\frac{(3 - a) \cdot 5}{2 (a - 3)}$

Cancel the common factor of $3 - a$ and $a - 3$ .

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{(- 1 (- 3) - a) \cdot 5}{2 (a - 3)}$

Factor $- 1$ out of $- a$ .

$\frac{(- 1 (- 3) - (a)) \cdot 5}{2 (a - 3)}$

Factor $- 1$ out of $- 1 (- 3) - (a)$ .

$\frac{- 1 (- 3 + a) \cdot 5}{2 (a - 3)}$

Reorder terms.

$\frac{- 1 (a - 3) \cdot 5}{2 (a - 3)}$

Cancel the common factor.

$\frac{- 1 (a - 3) \cdot 5}{2 (a - 3)}$

Rewrite the expression.

$\frac{- 1 \cdot 5}{2}$

Simplify the expression.

Multiply $- 1$ by $5$ .

$\frac{- 5}{2}$

Move the negative in front of the fraction.

$- \frac{5}{2}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{5}{2}$

Decimal Form:

$- 2.5$

Mixed Number Form:

$- 2 \frac{1}{2}$

Simplify ((9-a^2)/(a^2+5a+6))÷((2a-6)/(5a+10))

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