Simplify (6x^4-216x^2)/(x^2-5x-6)*(x^2+10x+9)/(3x^2+45x+162)

$\frac{6 x^{4} - 216 x^{2}}{x^{2} - 5 x - 6} \cdot \frac{x^{2} + 10 x + 9}{3 x^{2} + 45 x + 162}$

Simplify the numerator.

Factor $6 x^{2}$ out of $6 x^{4} - 216 x^{2}$ .

Factor $6 x^{2}$ out of $6 x^{4}$ .

$\frac{6 x^{2} (x^{2}) - 216 x^{2}}{x^{2} - 5 x - 6} \cdot \frac{x^{2} + 10 x + 9}{3 x^{2} + 45 x + 162}$

Factor $6 x^{2}$ out of $- 216 x^{2}$ .

$\frac{6 x^{2} (x^{2}) + 6 x^{2} (- 36)}{x^{2} - 5 x - 6} \cdot \frac{x^{2} + 10 x + 9}{3 x^{2} + 45 x + 162}$

Factor $6 x^{2}$ out of $6 x^{2} (x^{2}) + 6 x^{2} (- 36)$ .

$\frac{6 x^{2} (x^{2} - 36)}{x^{2} - 5 x - 6} \cdot \frac{x^{2} + 10 x + 9}{3 x^{2} + 45 x + 162}$

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

$\frac{6 x^{2} (x^{2} - 6^{2})}{x^{2} - 5 x - 6} \cdot \frac{x^{2} + 10 x + 9}{3 x^{2} + 45 x + 162}$

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 $b = 6$ .

$\frac{6 x^{2} (x + 6) (x - 6)}{x^{2} - 5 x - 6} \cdot \frac{x^{2} + 10 x + 9}{3 x^{2} + 45 x + 162}$

Factor $x^{2} - 5 x - 6$ 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 $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Write the factored form using these integers.

$\frac{6 x^{2} (x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{x^{2} + 10 x + 9}{3 x^{2} + 45 x + 162}$

Factor $x^{2} + 10 x + 9$ 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 $9$ and whose sum is $10$ .

$1, 9$

Write the factored form using these integers.

$\frac{6 x^{2} (x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{(x + 1) (x + 9)}{3 x^{2} + 45 x + 162}$

Simplify the denominator.

Factor $3$ out of $3 x^{2} + 45 x + 162$ .

Factor $3$ out of $3 x^{2}$ .

$\frac{6 x^{2} (x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{(x + 1) (x + 9)}{3 (x^{2}) + 45 x + 162}$

Factor $3$ out of $45 x$ .

$\frac{6 x^{2} (x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{(x + 1) (x + 9)}{3 (x^{2}) + 3 (15 x) + 162}$

Factor $3$ out of $162$ .

$\frac{6 x^{2} (x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{(x + 1) (x + 9)}{3 x^{2} + 3 (15 x) + 3 \cdot 54}$

Factor $3$ out of $3 x^{2} + 3 (15 x)$ .

$\frac{6 x^{2} (x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{(x + 1) (x + 9)}{3 (x^{2} + 15 x) + 3 \cdot 54}$

Factor $3$ out of $3 (x^{2} + 15 x) + 3 \cdot 54$ .

$\frac{6 x^{2} (x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{(x + 1) (x + 9)}{3 (x^{2} + 15 x + 54)}$

Factor $x^{2} + 15 x + 54$ 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 $54$ and whose sum is $15$ .

$6, 9$

Write the factored form using these integers.

$\frac{6 x^{2} (x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{(x + 1) (x + 9)}{3 ((x + 6) (x + 9))}$

$\frac{6 x^{2} (x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{(x + 1) (x + 9)}{3 (x + 6) (x + 9)}$

Simplify terms.

Combine.

$\frac{6 x^{2} (x + 6) (x - 6) ((x + 1) (x + 9))}{(x - 6) (x + 1) (3 (x + 6) (x + 9))}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6 x^{2} (x + 6) (x - 6) ((x + 1) (x + 9))$ .

$\frac{3 (2 x^{2} (x + 6) (x - 6) ((x + 1) (x + 9)))}{(x - 6) (x + 1) (3 (x + 6) (x + 9))}$

Cancel the common factors.

Factor $3$ out of $(x - 6) (x + 1) (3 (x + 6) (x + 9))$ .

$\frac{3 (2 x^{2} (x + 6) (x - 6) ((x + 1) (x + 9)))}{3 ((x - 6) (x + 1) ((x + 6) (x + 9)))}$

Cancel the common factor.

$\frac{3 (2 x^{2} (x + 6) (x - 6) ((x + 1) (x + 9)))}{3 ((x - 6) (x + 1) ((x + 6) (x + 9)))}$

Rewrite the expression.

$\frac{2 x^{2} (x + 6) (x - 6) ((x + 1) (x + 9))}{(x - 6) (x + 1) ((x + 6) (x + 9))}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$\frac{2 x^{2} (x + 6) (x - 6) ((x + 1) (x + 9))}{(x - 6) (x + 1) ((x + 6) (x + 9))}$

Rewrite the expression.

$\frac{(2 x^{2} (x - 6)) ((x + 1) (x + 9))}{(x - 6) (x + 1) (x + 9)}$

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{2 x^{2} (x - 6) ((x + 1) (x + 9))}{(x - 6) (x + 1) (x + 9)}$

Rewrite the expression.

$\frac{(2 x^{2}) ((x + 1) (x + 9))}{(x + 1) (x + 9)}$

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$\frac{2 x^{2} ((x + 1) (x + 9))}{(x + 1) (x + 9)}$

Rewrite the expression.

$\frac{(2 x^{2}) (x + 9)}{x + 9}$

Cancel the common factor of $x + 9$ .

Cancel the common factor.

$\frac{2 x^{2} (x + 9)}{x + 9}$

Divide $2 x^{2}$ by $1$ .

$2 x^{2}$

Simplify (6x^4-216x^2)/(x^2-5x-6)*(x^2+10x+9)/(3x^2+45x+162)

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