Simplify (x^2+2x-24)/(x^3+6x^2)*(x^2+3x)/(x^2-x-12)

$\frac{x^{2} + 2 x - 24}{x^{3} + 6 x^{2}} \cdot \frac{x^{2} + 3 x}{x^{2} - x - 12}$

Step 1

Factor $x^{2} + 2 x - 24$ 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 $- 24$ and whose sum is $2$ .

$- 4, 6$

Write the factored form using these integers.

$\frac{(x - 4) (x + 6)}{x^{3} + 6 x^{2}} \cdot \frac{x^{2} + 3 x}{x^{2} - x - 12}$

Step 2

Simplify with factoring out.

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

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

$\frac{(x - 4) (x + 6)}{x^{2} x + 6 x^{2}} \cdot \frac{x^{2} + 3 x}{x^{2} - x - 12}$

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

$\frac{(x - 4) (x + 6)}{x^{2} x + x^{2} \cdot 6} \cdot \frac{x^{2} + 3 x}{x^{2} - x - 12}$

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

$\frac{(x - 4) (x + 6)}{x^{2} (x + 6)} \cdot \frac{x^{2} + 3 x}{x^{2} - x - 12}$

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

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

$\frac{(x - 4) (x + 6)}{x^{2} (x + 6)} \cdot \frac{x \cdot x + 3 x}{x^{2} - x - 12}$

Factor $x$ out of $3 x$ .

$\frac{(x - 4) (x + 6)}{x^{2} (x + 6)} \cdot \frac{x \cdot x + x \cdot 3}{x^{2} - x - 12}$

Factor $x$ out of $x \cdot x + x \cdot 3$ .

$\frac{(x - 4) (x + 6)}{x^{2} (x + 6)} \cdot \frac{x (x + 3)}{x^{2} - x - 12}$

Step 3

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.

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

Step 4

Simplify terms.

Combine.

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

Cancel the common factor of $x - 4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 6$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Cancel the common factors.

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

$\frac{x (x + 3)}{x (x (x + 3))}$

Cancel the common factor.

$\frac{x (x + 3)}{x (x (x + 3))}$

Rewrite the expression.

$\frac{x + 3}{x (x + 3)}$

Step 6

Cancel the common factor of $x + 3$ .

Cancel the common factor.

$\frac{x + 3}{x (x + 3)}$

Rewrite the expression.

$\frac{1}{x}$

Simplify (x^2+2x-24)/(x^3+6x^2)*(x^2+3x)/(x^2-x-12)

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