Simplify (5x-1)/(x^2+2x-8)-6/(x+4)

$\frac{5 x - 1}{x^{2} + 2 x - 8} - \frac{6}{x + 4}$

Step 1

Factor $x^{2} + 2 x - 8$ 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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Write the factored form using these integers.

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

Step 2

To write $- \frac{6}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 3

Write each expression with a common denominator of $(x - 2) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{6}{x + 4}$ by $\frac{x - 2}{x - 2}$ .

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

Reorder the factors of $(x - 2) (x + 4)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 6$ by $- 2$ .

$\frac{5 x - 1 - 6 x + 12}{(x + 4) (x - 2)}$

Subtract $6 x$ from $5 x$ .

$\frac{- x - 1 + 12}{(x + 4) (x - 2)}$

Add $- 1$ and $12$ .

$\frac{- x + 11}{(x + 4) (x - 2)}$

Step 6

Simplify with factoring out.

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

$\frac{- (x) + 11}{(x + 4) (x - 2)}$

Rewrite $11$ as $- 1 (- 11)$ .

$\frac{- (x) - 1 (- 11)}{(x + 4) (x - 2)}$

Factor $- 1$ out of $- (x) - 1 (- 11)$ .

$\frac{- (x - 11)}{(x + 4) (x - 2)}$

Simplify the expression.

Rewrite $- (x - 11)$ as $- 1 (x - 11)$ .

$\frac{- 1 (x - 11)}{(x + 4) (x - 2)}$

Move the negative in front of the fraction.

$- \frac{x - 11}{(x + 4) (x - 2)}$

Simplify (5x-1)/(x^2+2x-8)-6/(x+4)

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