Simplify ((x-3)(x+6))/((x^2-7x+12)(x^2-36))

$\frac{(x - 3) (x + 6)}{(x^{2} - 7 x + 12) (x^{2} - 36)}$

Simplify the denominator.

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.

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

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

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

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 $b = 6$ .

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

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 6$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify ((x-3)(x+6))/((x^2-7x+12)(x^2-36))

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