Find the Product (x^2-x-6)/(4x^3)*(2x^2+2x)/(x^2+5x+6)

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

Step 1

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Write the factored form using these integers.

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

Step 2

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

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

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

Factor $2 x$ out of $2 x$ .

$\frac{(x - 3) (x + 2)}{4 x^{3}} \cdot \frac{2 x (x) + 2 x (1)}{x^{2} + 5 x + 6}$

Factor $2 x$ out of $2 x (x) + 2 x (1)$ .

$\frac{(x - 3) (x + 2)}{4 x^{3}} \cdot \frac{2 x (x + 1)}{x^{2} + 5 x + 6}$

Step 3

Factor $x^{2} + 5 x + 6$ 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 $6$ and whose sum is $5$ .

$2, 3$

Write the factored form using these integers.

$\frac{(x - 3) (x + 2)}{4 x^{3}} \cdot \frac{2 x (x + 1)}{(x + 2) (x + 3)}$

Step 4

Simplify terms.

Combine.

$\frac{(x - 3) (x + 2) (2 x (x + 1))}{4 x^{3} ((x + 2) (x + 3))}$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{(x - 3) (x + 2) (2 x (x + 1))}{4 x^{3} ((x + 2) (x + 3))}$

Rewrite the expression.

$\frac{(x - 3) (2 x (x + 1))}{4 x^{3} (x + 3)}$

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

Factor $2$ out of $(x - 3) (2 x (x + 1))$ .

$\frac{2 ((x - 3) (x (x + 1)))}{4 x^{3} (x + 3)}$

Cancel the common factors.

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

$\frac{2 ((x - 3) (x (x + 1)))}{2 (2 x^{3} (x + 3))}$

Cancel the common factor.

$\frac{2 ((x - 3) (x (x + 1)))}{2 (2 x^{3} (x + 3))}$

Rewrite the expression.

$\frac{(x - 3) (x (x + 1))}{2 x^{3} (x + 3)}$

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

Factor $x$ out of $(x - 3) (x (x + 1))$ .

$\frac{x ((x - 3) (x + 1))}{2 x^{3} (x + 3)}$

Cancel the common factors.

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

$\frac{x ((x - 3) (x + 1))}{x (2 x^{2} (x + 3))}$

Cancel the common factor.

$\frac{x ((x - 3) (x + 1))}{x (2 x^{2} (x + 3))}$

Rewrite the expression.

$\frac{(x - 3) (x + 1)}{2 x^{2} (x + 3)}$

Find the Product (x^2-x-6)/(4x^3)*(2x^2+2x)/(x^2+5x+6)

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