Simplify (x^3-4x)/(4x^2(x+10)^2)*(2x^2+8x-120)/(x^2-8x+12)

$\frac{x^{3} - 4 x}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 x^{2} + 8 x - 120}{x^{2} - 8 x + 12}$

Simplify the numerator.

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

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

$\frac{x \cdot x^{2} - 4 x}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 x^{2} + 8 x - 120}{x^{2} - 8 x + 12}$

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

$\frac{x \cdot x^{2} + x \cdot - 4}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 x^{2} + 8 x - 120}{x^{2} - 8 x + 12}$

Factor $x$ out of $x \cdot x^{2} + x \cdot - 4$ .

$\frac{x (x^{2} - 4)}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 x^{2} + 8 x - 120}{x^{2} - 8 x + 12}$

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

$\frac{x (x^{2} - 2^{2})}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 x^{2} + 8 x - 120}{x^{2} - 8 x + 12}$

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 $b = 2$ .

$\frac{x (x + 2) (x - 2)}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 x^{2} + 8 x - 120}{x^{2} - 8 x + 12}$

Simplify the numerator.

Factor $2$ out of $2 x^{2} + 8 x - 120$ .

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

$\frac{x (x + 2) (x - 2)}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 (x^{2}) + 8 x - 120}{x^{2} - 8 x + 12}$

Factor $2$ out of $8 x$ .

$\frac{x (x + 2) (x - 2)}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 (x^{2}) + 2 (4 x) - 120}{x^{2} - 8 x + 12}$

Factor $2$ out of $- 120$ .

$\frac{x (x + 2) (x - 2)}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 x^{2} + 2 (4 x) + 2 \cdot - 60}{x^{2} - 8 x + 12}$

Factor $2$ out of $2 x^{2} + 2 (4 x)$ .

$\frac{x (x + 2) (x - 2)}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 (x^{2} + 4 x) + 2 \cdot - 60}{x^{2} - 8 x + 12}$

Factor $2$ out of $2 (x^{2} + 4 x) + 2 \cdot - 60$ .

$\frac{x (x + 2) (x - 2)}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 (x^{2} + 4 x - 60)}{x^{2} - 8 x + 12}$

Factor $x^{2} + 4 x - 60$ 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 $- 60$ and whose sum is $4$ .

$- 6, 10$

Write the factored form using these integers.

$\frac{x (x + 2) (x - 2)}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 ((x - 6) (x + 10))}{x^{2} - 8 x + 12}$

$\frac{x (x + 2) (x - 2)}{4 x^{2} {(x + 10)}^{2}} \cdot \frac{2 (x - 6) (x + 10)}{x^{2} - 8 x + 12}$

Factor $x^{2} - 8 x + 12$ 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 $- 8$ .

$- 6, - 2$

Write the factored form using these integers.

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

Simplify terms.

Combine.

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

Cancel the common factor of $x$ and $x^{2}$ .

Factor $x$ out of $x (x + 2) (x - 2) (2 (x - 6) (x + 10))$ .

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

Cancel the common factors.

Factor $x$ out of $4 x^{2} {(x + 10)}^{2} ((x - 6) (x - 2))$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $(x + 2) (2 (x - 6) (x + 10))$ .

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

Cancel the common factors.

Factor $2$ out of $4 x {(x + 10)}^{2} (x - 6)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 6$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{(x + 2) (x + 10)}{2 x {(x + 10)}^{2}}$

Cancel the common factor of $x + 10$ and ${(x + 10)}^{2}$ .

Factor $x + 10$ out of $(x + 2) (x + 10)$ .

$\frac{(x + 10) (x + 2)}{2 x {(x + 10)}^{2}}$

Cancel the common factors.

Factor $x + 10$ out of $2 x {(x + 10)}^{2}$ .

$\frac{(x + 10) (x + 2)}{(x + 10) (2 x (x + 10))}$

Cancel the common factor.

$\frac{(x + 10) (x + 2)}{(x + 10) (2 x (x + 10))}$

Rewrite the expression.

$\frac{x + 2}{2 x (x + 10)}$

Simplify (x^3-4x)/(4x^2(x+10)^2)*(2x^2+8x-120)/(x^2-8x+12)

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