Simplify (x^2-2x-15)/(x^2+x-12)*(2x^2-6x)/(x^3+3x^2)

$\frac{x^{2} - 2 x - 15}{x^{2} + x - 12} \cdot \frac{2 x^{2} - 6 x}{x^{3} + 3 x^{2}}$

Step 1

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

$- 5, 3$

Write the factored form using these integers.

$\frac{(x - 5) (x + 3)}{x^{2} + x - 12} \cdot \frac{2 x^{2} - 6 x}{x^{3} + 3 x^{2}}$

Step 2

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 $1$ .

$- 3, 4$

Write the factored form using these integers.

$\frac{(x - 5) (x + 3)}{(x - 3) (x + 4)} \cdot \frac{2 x^{2} - 6 x}{x^{3} + 3 x^{2}}$

Step 3

Simplify terms.

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

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

$\frac{(x - 5) (x + 3)}{(x - 3) (x + 4)} \cdot \frac{2 x (x) - 6 x}{x^{3} + 3 x^{2}}$

Factor $2 x$ out of $- 6 x$ .

$\frac{(x - 5) (x + 3)}{(x - 3) (x + 4)} \cdot \frac{2 x (x) + 2 x (- 3)}{x^{3} + 3 x^{2}}$

Factor $2 x$ out of $2 x (x) + 2 x (- 3)$ .

$\frac{(x - 5) (x + 3)}{(x - 3) (x + 4)} \cdot \frac{2 x (x - 3)}{x^{3} + 3 x^{2}}$

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

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

$\frac{(x - 5) (x + 3)}{(x - 3) (x + 4)} \cdot \frac{2 x (x - 3)}{x^{2} x + 3 x^{2}}$

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

$\frac{(x - 5) (x + 3)}{(x - 3) (x + 4)} \cdot \frac{2 x (x - 3)}{x^{2} x + x^{2} \cdot 3}$

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

$\frac{(x - 5) (x + 3)}{(x - 3) (x + 4)} \cdot \frac{2 x (x - 3)}{x^{2} (x + 3)}$

Combine.

$\frac{(x - 5) (x + 3) (2 x (x - 3))}{(x - 3) (x + 4) (x^{2} (x + 3))}$

Cancel the common factor of $x + 3$ .

Cancel the common factor.

$\frac{(x - 5) (x + 3) (2 x (x - 3))}{(x - 3) (x + 4) (x^{2} (x + 3))}$

Rewrite the expression.

$\frac{(x - 5) (2 x (x - 3))}{(x - 3) (x + 4) (x^{2})}$

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

Factor $x$ out of $(x - 5) (2 x (x - 3))$ .

$\frac{x ((x - 5) (2 (x - 3)))}{(x - 3) (x + 4) (x^{2})}$

Cancel the common factors.

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

$\frac{x ((x - 5) (2 (x - 3)))}{x ((x - 3) (x + 4) (x))}$

Cancel the common factor.

$\frac{x ((x - 5) (2 (x - 3)))}{x ((x - 3) (x + 4) (x))}$

Rewrite the expression.

$\frac{(x - 5) (2 (x - 3))}{(x - 3) (x + 4) (x)}$

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{(x - 5) (2 (x - 3))}{(x - 3) (x + 4) x}$

Rewrite the expression.

$\frac{(x - 5) \cdot (2)}{(x + 4) (x)}$

Simplify the expression.

Move $2$ to the left of $x - 5$ .

$\frac{2 \cdot (x - 5)}{(x + 4) (x)}$

Reorder factors in $\frac{2 (x - 5)}{(x + 4) (x)}$ .

$\frac{2 (x - 5)}{x (x + 4)}$

Simplify (x^2-2x-15)/(x^2+x-12)*(2x^2-6x)/(x^3+3x^2)

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