Simplify ((x^2-x-6)/(2x^4-6x^3))÷((x+2)/(4x^3))

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

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.

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

Step 3

Simplify terms.

Factor $2 x^{3}$ out of $2 x^{4} - 6 x^{3}$ .

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

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

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

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

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

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

Cancel the common factor of $x + 2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2 x^{3}$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 3}{x - 3} \cdot 2$

Combine $\frac{x - 3}{x - 3}$ and $2$ .

$\frac{(x - 3) \cdot 2}{x - 3}$

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{(x - 3) \cdot 2}{x - 3}$

Divide $2$ by $1$ .

$2$

Simplify ((x^2-x-6)/(2x^4-6x^3))÷((x+2)/(4x^3))

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