Simplify (x+2)/(x^2-3x-18)-(5x)/(x-6)

$\frac{x + 2}{x^{2} - 3 x - 18} - \frac{5 x}{x - 6}$

Factor $x^{2} - 3 x - 18$ 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 $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Write the factored form using these integers.

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

To write $- \frac{5 x}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

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

Simplify terms.

Multiply $\frac{5 x}{x - 6}$ by $\frac{x + 3}{x + 3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $- 5$ .

$\frac{x + 2 - 5 x^{2} - 15 x}{(x - 6) (x + 3)}$

Subtract $15 x$ from $x$ .

$\frac{- 14 x + 2 - 5 x^{2}}{(x - 6) (x + 3)}$

Reorder terms.

$\frac{- 5 x^{2} - 14 x + 2}{(x - 6) (x + 3)}$

Simplify with factoring out.

Factor $- 1$ out of $- 5 x^{2}$ .

$\frac{- (5 x^{2}) - 14 x + 2}{(x - 6) (x + 3)}$

Factor $- 1$ out of $- 14 x$ .

$\frac{- (5 x^{2}) - (14 x) + 2}{(x - 6) (x + 3)}$

Factor $- 1$ out of $- (5 x^{2}) - (14 x)$ .

$\frac{- (5 x^{2} + 14 x) + 2}{(x - 6) (x + 3)}$

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- (5 x^{2} + 14 x) - 1 (- 2)}{(x - 6) (x + 3)}$

Factor $- 1$ out of $- (5 x^{2} + 14 x) - 1 (- 2)$ .

$\frac{- (5 x^{2} + 14 x - 2)}{(x - 6) (x + 3)}$

Simplify the expression.

Rewrite $- (5 x^{2} + 14 x - 2)$ as $- 1 (5 x^{2} + 14 x - 2)$ .

$\frac{- 1 (5 x^{2} + 14 x - 2)}{(x - 6) (x + 3)}$

Move the negative in front of the fraction.

$- \frac{5 x^{2} + 14 x - 2}{(x - 6) (x + 3)}$

Simplify (x+2)/(x^2-3x-18)-(5x)/(x-6)

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