Simplify ((x^2-7x+12)/(x^2+x-12)*(-4x)/(-x-2))÷((x-4)/(x+4))

$\frac{x^{2} - 7 x + 12}{x^{2} + x - 12} \cdot \frac{- 4 x}{- x - 2} \div \frac{x - 4}{x + 4}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 7 x + 12}{x^{2} + x - 12} \cdot \frac{- 4 x}{- x - 2} \frac{x + 4}{x - 4}$

Factor $x^{2} - 7 x + 12$ 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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Write the factored form using these integers.

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

Factor $x^{2} + x - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Write the factored form using these integers.

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

Simplify terms.

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 4}{x + 4} \cdot \frac{- 4 x}{- x - 2} \frac{x + 4}{x - 4}$

Simplify the expression.

Move the negative in front of the fraction.

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

Rewrite using the commutative property of multiplication.

$- \frac{x - 4}{x + 4} \cdot \frac{4 x}{- x - 2} \frac{x + 4}{x - 4}$

Multiply $\frac{4 x}{- x - 2}$ by $\frac{x - 4}{x + 4}$ .

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

Cancel the common factor of $x - 4$ .

Move the leading negative in $- \frac{4 x (x - 4)}{(- x - 2) (x + 4)}$ into the numerator.

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 4$ .

Factor $x + 4$ out of $(- x - 2) (x + 4)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{- 4 x}{- x - 2}$

Move the negative in front of the fraction.

$- \frac{(4) x}{- x - 2}$

Factor $- 1$ out of $- x$ .

$- \frac{4 x}{- (x) - 2}$

Rewrite $- 2$ as $- 1 (2)$ .

$- \frac{4 x}{- (x) - 1 (2)}$

Factor $- 1$ out of $- (x) - 1 (2)$ .

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

Simplify the expression.

Rewrite $- (x + 2)$ as $- 1 (x + 2)$ .

$- \frac{4 x}{- 1 (x + 2)}$

Move the negative in front of the fraction.

$- - \frac{4 x}{x + 2}$

Multiply $- 1$ by $- 1$ .

$1 \frac{4 x}{x + 2}$

Multiply $\frac{4 x}{x + 2}$ by $1$ .

$\frac{4 x}{x + 2}$

Simplify ((x^2-7x+12)/(x^2+x-12)*(-4x)/(-x-2))÷((x-4)/(x+4))

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