Divide ((3x)/(x^2+10x+24))÷((x-6)/(x^2-36))

$\frac{3 x}{x^{2} + 10 x + 24} \div \frac{x - 6}{x^{2} - 36}$

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x}{x^{2} + 10 x + 24} \cdot \frac{x^{2} - 36}{x - 6}$

Factor $x^{2} + 10 x + 24$ 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 $24$ and whose sum is $10$ .

$4, 6$

Write the factored form using these integers.

$\frac{3 x}{(x + 4) (x + 6)} \cdot \frac{x^{2} - 36}{x - 6}$

Simplify the numerator.

Rewrite $36$ as $6^{2}$ .

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

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 $b = 6$ .

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

Cancel the common factor of $x + 6$ .

Factor $x + 6$ out of $(x + 4) (x + 6)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{3 x}{x + 4}$ by $\frac{x - 6}{x - 6}$ .

$\frac{3 x (x - 6)}{(x + 4) (x - 6)}$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{3 x (x - 6)}{(x + 4) (x - 6)}$

Rewrite the expression.

$\frac{3 x}{x + 4}$

Divide ((3x)/(x^2+10x+24))÷((x-6)/(x^2-36))

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