Simplify (-2x^2-9x)/(4x^2-81)

$\frac{- 2 x^{2} - 9 x}{4 x^{2} - 81}$

Step 1

Factor $x$ out of $- 2 x^{2} - 9 x$ .

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

$\frac{x (- 2 x) - 9 x}{4 x^{2} - 81}$

Factor $x$ out of $- 9 x$ .

$\frac{x (- 2 x) + x \cdot - 9}{4 x^{2} - 81}$

Factor $x$ out of $x (- 2 x) + x \cdot - 9$ .

$\frac{x (- 2 x - 9)}{4 x^{2} - 81}$

Step 2

Simplify the denominator.

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{x (- 2 x - 9)}{{(2 x)}^{2} - 81}$

Rewrite $81$ as $9^{2}$ .

$\frac{x (- 2 x - 9)}{{(2 x)}^{2} - 9^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 9$ .

$\frac{x (- 2 x - 9)}{(2 x + 9) (2 x - 9)}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 2 x - 9$ and $2 x + 9$ .

Factor $- 1$ out of $- 2 x$ .

$\frac{x (- (2 x) - 9)}{(2 x + 9) (2 x - 9)}$

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{x (- (2 x) - 1 (9))}{(2 x + 9) (2 x - 9)}$

Factor $- 1$ out of $- (2 x) - 1 (9)$ .

$\frac{x (- (2 x + 9))}{(2 x + 9) (2 x - 9)}$

Rewrite $- (2 x + 9)$ as $- 1 (2 x + 9)$ .

$\frac{x (- 1 (2 x + 9))}{(2 x + 9) (2 x - 9)}$

Cancel the common factor.

$\frac{x (- 1 (2 x + 9))}{(2 x + 9) (2 x - 9)}$

Rewrite the expression.

$\frac{x \cdot (- 1)}{2 x - 9}$

Simplify the expression.

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{2 x - 9}$

Move the negative in front of the fraction.

$- \frac{x}{2 x - 9}$