Divide ((x^2-6x+8)/(3x-12))÷((x^2-4)/(x^2+5x+6))

$\frac{x^{2} - 6 x + 8}{3 x - 12} \div \frac{x^{2} - 4}{x^{2} + 5 x + 6}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 6 x + 8}{3 x - 12} \cdot \frac{x^{2} + 5 x + 6}{x^{2} - 4}$

Factor $x^{2} - 6 x + 8$ 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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Write the factored form using these integers.

$\frac{(x - 4) (x - 2)}{3 x - 12} \cdot \frac{x^{2} + 5 x + 6}{x^{2} - 4}$

Factor $3$ out of $3 x - 12$ .

Factor $3$ out of $3 x$ .

$\frac{(x - 4) (x - 2)}{3 (x) - 12} \cdot \frac{x^{2} + 5 x + 6}{x^{2} - 4}$

Factor $3$ out of $- 12$ .

$\frac{(x - 4) (x - 2)}{3 x + 3 \cdot - 4} \cdot \frac{x^{2} + 5 x + 6}{x^{2} - 4}$

Factor $3$ out of $3 x + 3 \cdot - 4$ .

$\frac{(x - 4) (x - 2)}{3 (x - 4)} \cdot \frac{x^{2} + 5 x + 6}{x^{2} - 4}$

Cancel the common factor of $x - 4$ .

Cancel the common factor.

$\frac{(x - 4) (x - 2)}{3 (x - 4)} \cdot \frac{x^{2} + 5 x + 6}{x^{2} - 4}$

Rewrite the expression.

$\frac{x - 2}{3} \cdot \frac{x^{2} + 5 x + 6}{x^{2} - 4}$

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.

$\frac{x - 2}{3} \cdot \frac{(x + 2) (x + 3)}{x^{2} - 4}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

Cancel the common factor of $x - 2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{1}{3}$ by $\frac{(x + 2) (x + 3)}{x + 2}$ .

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

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Split the fraction $\frac{x + 3}{3}$ into two fractions.

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

Divide $3$ by $3$ .

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

Divide ((x^2-6x+8)/(3x-12))÷((x^2-4)/(x^2+5x+6))

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