Simplify ((4x-24)/(-3x-15))/((x^2-36)/(x+5))

$\frac{\frac{4 x - 24}{- 3 x - 15}}{\frac{x^{2} - 36}{x + 5}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 x - 24}{- 3 x - 15} \cdot \frac{x + 5}{x^{2} - 36}$

Step 2

Simplify with factoring out.

Factor $4$ out of $4 x - 24$ .

Factor $4$ out of $4 x$ .

$\frac{4 (x) - 24}{- 3 x - 15} \cdot \frac{x + 5}{x^{2} - 36}$

Factor $4$ out of $- 24$ .

$\frac{4 x + 4 \cdot - 6}{- 3 x - 15} \cdot \frac{x + 5}{x^{2} - 36}$

Factor $4$ out of $4 x + 4 \cdot - 6$ .

$\frac{4 (x - 6)}{- 3 x - 15} \cdot \frac{x + 5}{x^{2} - 36}$

Factor $3$ out of $- 3 x - 15$ .

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

$\frac{4 (x - 6)}{3 (- x) - 15} \cdot \frac{x + 5}{x^{2} - 36}$

Factor $3$ out of $- 15$ .

$\frac{4 (x - 6)}{3 (- x) + 3 (- 5)} \cdot \frac{x + 5}{x^{2} - 36}$

Factor $3$ out of $3 (- x) + 3 (- 5)$ .

$\frac{4 (x - 6)}{3 (- x - 5)} \cdot \frac{x + 5}{x^{2} - 36}$

Step 3

Simplify the denominator.

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

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

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 $b = 6$ .

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

Step 4

Simplify terms.

Cancel the common factor of $x - 6$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{4}{3 (- x - 5)}$ by $\frac{x + 5}{x + 6}$ .

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

Cancel the common factor of $x + 5$ and $- x - 5$ .

Factor $- 1$ out of $x$ .

$\frac{4 (- 1 (- x) + 5)}{3 (- x - 5) (x + 6)}$

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{4 (- 1 (- x) - 1 (- 5))}{3 (- x - 5) (x + 6)}$

Factor $- 1$ out of $- 1 (- x) - 1 (- 5)$ .

$\frac{4 (- 1 (- x - 5))}{3 (- x - 5) (x + 6)}$

Cancel the common factor.

$\frac{4 (- 1 (- x - 5))}{3 (- x - 5) (x + 6)}$

Rewrite the expression.

$\frac{4 \cdot (- 1)}{3 (x + 6)}$

Step 5

Multiply $4$ by $- 1$ .

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

Step 6

Move the negative in front of the fraction.

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

Simplify ((4x-24)/(-3x-15))/((x^2-36)/(x+5))

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