Simplify (x^3+3x^2)/(x^3-6x^2-40x)*(x^2-16)/(x^2-x-12)

$\frac{x^{3} + 3 x^{2}}{x^{3} - 6 x^{2} - 40 x} \cdot \frac{x^{2} - 16}{x^{2} - x - 12}$

Step 1

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

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

$\frac{x^{2} x + 3 x^{2}}{x^{3} - 6 x^{2} - 40 x} \cdot \frac{x^{2} - 16}{x^{2} - x - 12}$

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

$\frac{x^{2} x + x^{2} \cdot 3}{x^{3} - 6 x^{2} - 40 x} \cdot \frac{x^{2} - 16}{x^{2} - x - 12}$

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 3$ .

$\frac{x^{2} (x + 3)}{x^{3} - 6 x^{2} - 40 x} \cdot \frac{x^{2} - 16}{x^{2} - x - 12}$

Step 2

Simplify the denominator.

Factor $x$ out of $x^{3} - 6 x^{2} - 40 x$ .

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

$\frac{x^{2} (x + 3)}{x \cdot x^{2} - 6 x^{2} - 40 x} \cdot \frac{x^{2} - 16}{x^{2} - x - 12}$

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

$\frac{x^{2} (x + 3)}{x \cdot x^{2} + x (- 6 x) - 40 x} \cdot \frac{x^{2} - 16}{x^{2} - x - 12}$

Factor $x$ out of $- 40 x$ .

$\frac{x^{2} (x + 3)}{x \cdot x^{2} + x (- 6 x) + x \cdot - 40} \cdot \frac{x^{2} - 16}{x^{2} - x - 12}$

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

$\frac{x^{2} (x + 3)}{x (x^{2} - 6 x) + x \cdot - 40} \cdot \frac{x^{2} - 16}{x^{2} - x - 12}$

Factor $x$ out of $x (x^{2} - 6 x) + x \cdot - 40$ .

$\frac{x^{2} (x + 3)}{x (x^{2} - 6 x - 40)} \cdot \frac{x^{2} - 16}{x^{2} - x - 12}$

Factor $x^{2} - 6 x - 40$ 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 $- 6$ .

$- 10, 4$

Write the factored form using these integers.

$\frac{x^{2} (x + 3)}{x ((x - 10) (x + 4))} \cdot \frac{x^{2} - 16}{x^{2} - x - 12}$

$\frac{x^{2} (x + 3)}{x (x - 10) (x + 4)} \cdot \frac{x^{2} - 16}{x^{2} - x - 12}$

Step 3

Simplify the numerator.

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

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

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 $b = 4$ .

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

Step 4

Factor $x^{2} - x - 12$ 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 $- 1$ .

$- 4, 3$

Write the factored form using these integers.

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

Step 5

Simplify terms.

Combine.

$\frac{x^{2} (x + 3) ((x + 4) (x - 4))}{x (x - 10) (x + 4) ((x - 4) (x + 3))}$

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

Factor $x$ out of $x^{2} (x + 3) ((x + 4) (x - 4))$ .

$\frac{x ((x (x + 3)) ((x + 4) (x - 4)))}{x (x - 10) (x + 4) ((x - 4) (x + 3))}$

Cancel the common factors.

Factor $x$ out of $x (x - 10) (x + 4) ((x - 4) (x + 3))$ .

$\frac{x ((x (x + 3)) ((x + 4) (x - 4)))}{x (((x - 10) (x + 4)) ((x - 4) (x + 3)))}$

Cancel the common factor.

$\frac{x ((x (x + 3)) ((x + 4) (x - 4)))}{x (((x - 10) (x + 4)) ((x - 4) (x + 3)))}$

Rewrite the expression.

$\frac{(x (x + 3)) ((x + 4) (x - 4))}{((x - 10) (x + 4)) ((x - 4) (x + 3))}$

Cancel the common factor of $x + 3$ .

Cancel the common factor.

$\frac{x (x + 3) ((x + 4) (x - 4))}{(x - 10) (x + 4) ((x - 4) (x + 3))}$

Rewrite the expression.

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

Cancel the common factor of $x + 4$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{(x) (x - 4)}{(x - 10) (x - 4)}$

Cancel the common factor of $x - 4$ .

Cancel the common factor.

$\frac{x (x - 4)}{(x - 10) (x - 4)}$

Rewrite the expression.

$\frac{x}{x - 10}$

Simplify (x^3+3x^2)/(x^3-6x^2-40x)*(x^2-16)/(x^2-x-12)

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