Simplify (x^3-16x)/(x^5+14x^4+40x^3)*(9x^2+18x-720)/(6x-24)

$\frac{x^{3} - 16 x}{x^{5} + 14 x^{4} + 40 x^{3}} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

Simplify the numerator.

Factor $x$ out of $x^{3} - 16 x$ .

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

$\frac{x \cdot x^{2} - 16 x}{x^{5} + 14 x^{4} + 40 x^{3}} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

Factor $x$ out of $- 16 x$ .

$\frac{x \cdot x^{2} + x \cdot - 16}{x^{5} + 14 x^{4} + 40 x^{3}} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

Factor $x$ out of $x \cdot x^{2} + x \cdot - 16$ .

$\frac{x (x^{2} - 16)}{x^{5} + 14 x^{4} + 40 x^{3}} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

Rewrite $16$ as $4^{2}$ .

$\frac{x (x^{2} - 4^{2})}{x^{5} + 14 x^{4} + 40 x^{3}} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

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 $b = 4$ .

$\frac{x (x + 4) (x - 4)}{x^{5} + 14 x^{4} + 40 x^{3}} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

Simplify the denominator.

Factor $x^{3}$ out of $x^{5} + 14 x^{4} + 40 x^{3}$ .

Factor $x^{3}$ out of $x^{5}$ .

$\frac{x (x + 4) (x - 4)}{x^{3} x^{2} + 14 x^{4} + 40 x^{3}} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

Factor $x^{3}$ out of $14 x^{4}$ .

$\frac{x (x + 4) (x - 4)}{x^{3} x^{2} + x^{3} (14 x) + 40 x^{3}} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

Factor $x^{3}$ out of $40 x^{3}$ .

$\frac{x (x + 4) (x - 4)}{x^{3} x^{2} + x^{3} (14 x) + x^{3} \cdot 40} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

Factor $x^{3}$ out of $x^{3} x^{2} + x^{3} (14 x)$ .

$\frac{x (x + 4) (x - 4)}{x^{3} (x^{2} + 14 x) + x^{3} \cdot 40} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

Factor $x^{3}$ out of $x^{3} (x^{2} + 14 x) + x^{3} \cdot 40$ .

$\frac{x (x + 4) (x - 4)}{x^{3} (x^{2} + 14 x + 40)} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

Factor $x^{2} + 14 x + 40$ 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 $40$ and whose sum is $14$ .

$4, 10$

Write the factored form using these integers.

$\frac{x (x + 4) (x - 4)}{x^{3} ((x + 4) (x + 10))} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

$\frac{x (x + 4) (x - 4)}{x^{3} (x + 4) (x + 10)} \cdot \frac{9 x^{2} + 18 x - 720}{6 x - 24}$

Simplify the numerator.

Factor $9$ out of $9 x^{2} + 18 x - 720$ .

Factor $9$ out of $9 x^{2}$ .

$\frac{x (x + 4) (x - 4)}{x^{3} (x + 4) (x + 10)} \cdot \frac{9 (x^{2}) + 18 x - 720}{6 x - 24}$

Factor $9$ out of $18 x$ .

$\frac{x (x + 4) (x - 4)}{x^{3} (x + 4) (x + 10)} \cdot \frac{9 (x^{2}) + 9 (2 x) - 720}{6 x - 24}$

Factor $9$ out of $- 720$ .

$\frac{x (x + 4) (x - 4)}{x^{3} (x + 4) (x + 10)} \cdot \frac{9 x^{2} + 9 (2 x) + 9 \cdot - 80}{6 x - 24}$

Factor $9$ out of $9 x^{2} + 9 (2 x)$ .

$\frac{x (x + 4) (x - 4)}{x^{3} (x + 4) (x + 10)} \cdot \frac{9 (x^{2} + 2 x) + 9 \cdot - 80}{6 x - 24}$

Factor $9$ out of $9 (x^{2} + 2 x) + 9 \cdot - 80$ .

$\frac{x (x + 4) (x - 4)}{x^{3} (x + 4) (x + 10)} \cdot \frac{9 (x^{2} + 2 x - 80)}{6 x - 24}$

Factor $x^{2} + 2 x - 80$ 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 $- 80$ and whose sum is $2$ .

$- 8, 10$

Write the factored form using these integers.

$\frac{x (x + 4) (x - 4)}{x^{3} (x + 4) (x + 10)} \cdot \frac{9 ((x - 8) (x + 10))}{6 x - 24}$

$\frac{x (x + 4) (x - 4)}{x^{3} (x + 4) (x + 10)} \cdot \frac{9 (x - 8) (x + 10)}{6 x - 24}$

Simplify terms.

Factor $6$ out of $6 x - 24$ .

Factor $6$ out of $6 x$ .

$\frac{x (x + 4) (x - 4)}{x^{3} (x + 4) (x + 10)} \cdot \frac{9 (x - 8) (x + 10)}{6 (x) - 24}$

Factor $6$ out of $- 24$ .

$\frac{x (x + 4) (x - 4)}{x^{3} (x + 4) (x + 10)} \cdot \frac{9 (x - 8) (x + 10)}{6 x + 6 \cdot - 4}$

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

$\frac{x (x + 4) (x - 4)}{x^{3} (x + 4) (x + 10)} \cdot \frac{9 (x - 8) (x + 10)}{6 (x - 4)}$

Combine.

$\frac{x (x + 4) (x - 4) (9 (x - 8) (x + 10))}{x^{3} (x + 4) (x + 10) (6 (x - 4))}$

Cancel the common factor of $x$ and $x^{3}$ .

Factor $x$ out of $x (x + 4) (x - 4) (9 (x - 8) (x + 10))$ .

$\frac{x (((x + 4) (x - 4)) (9 (x - 8) (x + 10)))}{x^{3} (x + 4) (x + 10) (6 (x - 4))}$

Cancel the common factors.

Factor $x$ out of $x^{3} (x + 4) (x + 10) (6 (x - 4))$ .

$\frac{x (((x + 4) (x - 4)) (9 (x - 8) (x + 10)))}{x (((x^{2} (x + 4)) (x + 10)) (6 (x - 4)))}$

Cancel the common factor.

$\frac{x (((x + 4) (x - 4)) (9 (x - 8) (x + 10)))}{x (((x^{2} (x + 4)) (x + 10)) (6 (x - 4)))}$

Rewrite the expression.

$\frac{((x + 4) (x - 4)) (9 (x - 8) (x + 10))}{((x^{2} (x + 4)) (x + 10)) (6 (x - 4))}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{(x + 4) (x - 4) (9 (x - 8) (x + 10))}{x^{2} (x + 4) (x + 10) (6 (x - 4))}$

Rewrite the expression.

$\frac{(x - 4) (9 (x - 8) (x + 10))}{((x^{2}) (x + 10)) (6 (x - 4))}$

Cancel the common factor of $x - 4$ .

Cancel the common factor.

$\frac{(x - 4) (9 (x - 8) (x + 10))}{(x^{2}) (x + 10) (6 (x - 4))}$

Rewrite the expression.

$\frac{9 (x - 8) (x + 10)}{((x^{2}) (x + 10)) \cdot (6)}$

Cancel the common factor of $9$ and $6$ .

Factor $3$ out of $9 (x - 8) (x + 10)$ .

$\frac{3 (3 (x - 8) (x + 10))}{((x^{2}) (x + 10)) \cdot (6)}$

Cancel the common factors.

Factor $3$ out of $((x^{2}) (x + 10)) \cdot (6)$ .

$\frac{3 (3 (x - 8) (x + 10))}{3 ((x^{2}) (x + 10) \cdot (2))}$

Cancel the common factor.

$\frac{3 (3 (x - 8) (x + 10))}{3 ((x^{2}) (x + 10) \cdot (2))}$

Rewrite the expression.

$\frac{3 (x - 8) (x + 10)}{(x^{2}) (x + 10) \cdot (2)}$

Cancel the common factor of $x + 10$ .

Cancel the common factor.

$\frac{3 (x - 8) (x + 10)}{x^{2} (x + 10) \cdot 2}$

Rewrite the expression.

$\frac{3 (x - 8)}{(x^{2}) \cdot (2)}$

Move $2$ to the left of $x^{2}$ .

$\frac{3 (x - 8)}{2 x^{2}}$

Simplify (x^3-16x)/(x^5+14x^4+40x^3)*(9x^2+18x-720)/(6x-24)

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