Simplify ((x-4)(x^2+2x-48))/((36-x^2)(x^2+4x-32))

$\frac{(x - 4) (x^{2} + 2 x - 48)}{(36 - x^{2}) (x^{2} + 4 x - 32)}$

Step 1

Factor $x^{2} + 2 x - 48$ 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 $- 48$ and whose sum is $2$ .

$- 6, 8$

Write the factored form using these integers.

$\frac{(x - 4) ((x - 6) (x + 8))}{(36 - x^{2}) (x^{2} + 4 x - 32)}$

$\frac{(x - 4) (x - 6) (x + 8)}{(36 - x^{2}) (x^{2} + 4 x - 32)}$

Step 2

Simplify the denominator.

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

$\frac{(x - 4) (x - 6) (x + 8)}{(6^{2} - x^{2}) (x^{2} + 4 x - 32)}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = x$ .

$\frac{(x - 4) (x - 6) (x + 8)}{(6 + x) (6 - x) (x^{2} + 4 x - 32)}$

Factor $x^{2} + 4 x - 32$ 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 $- 32$ and whose sum is $4$ .

$- 4, 8$

Write the factored form using these integers.

$\frac{(x - 4) (x - 6) (x + 8)}{(6 + x) (6 - x) ((x - 4) (x + 8))}$

$\frac{(x - 4) (x - 6) (x + 8)}{(6 + x) (6 - x) (x - 4) (x + 8)}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x - 4$ .

Cancel the common factor.

$\frac{(x - 4) (x - 6) (x + 8)}{(6 + x) (6 - x) (x - 4) (x + 8)}$

Rewrite the expression.

$\frac{(x - 6) (x + 8)}{(6 + x) (6 - x) (x + 8)}$

Cancel the common factor of $x + 8$ .

Cancel the common factor.

$\frac{(x - 6) (x + 8)}{(6 + x) (6 - x) (x + 8)}$

Rewrite the expression.

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

Cancel the common factor of $x - 6$ and $6 - x$ .

Factor $- 1$ out of $x$ .

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

Rewrite $- 6$ as $- 1 (6)$ .

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

Factor $- 1$ out of $- 1 (- x) - 1 (6)$ .

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{- 1}{6 + x}$

Move the negative in front of the fraction.

$- \frac{1}{6 + x}$

Simplify ((x-4)(x^2+2x-48))/((36-x^2)(x^2+4x-32))

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