Simplify 6/(x^2+4x-32)-(x-5)/(x-4)

$\frac{6}{x^{2} + 4 x - 32} - \frac{x - 5}{x - 4}$

Step 1

Factor $x^{2} + 4 x - 32$ 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 $- 32$ and whose sum is $4$ .

$- 4, 8$

Write the factored form using these integers.

$\frac{6}{(x - 4) (x + 8)} - \frac{x - 5}{x - 4}$

Step 2

To write $- \frac{x - 5}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x + 8}{x + 8}$ .

$\frac{6}{(x - 4) (x + 8)} - \frac{x - 5}{x - 4} \cdot \frac{x + 8}{x + 8}$

Step 3

Simplify terms.

Multiply $\frac{x - 5}{x - 4}$ by $\frac{x + 8}{x + 8}$ .

$\frac{6}{(x - 4) (x + 8)} - \frac{(x - 5) (x + 8)}{(x - 4) (x + 8)}$

Combine the numerators over the common denominator.

$\frac{6 - (x - 5) (x + 8)}{(x - 4) (x + 8)}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{6 + (- x - - 5) (x + 8)}{(x - 4) (x + 8)}$

Multiply $- 1$ by $- 5$ .

$\frac{6 + (- x + 5) (x + 8)}{(x - 4) (x + 8)}$

Expand $(- x + 5) (x + 8)$ using the FOIL Method.

Apply the distributive property.

$\frac{6 - x (x + 8) + 5 (x + 8)}{(x - 4) (x + 8)}$

Apply the distributive property.

$\frac{6 - x \cdot x - x \cdot 8 + 5 (x + 8)}{(x - 4) (x + 8)}$

Apply the distributive property.

$\frac{6 - x \cdot x - x \cdot 8 + 5 x + 5 \cdot 8}{(x - 4) (x + 8)}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{6 - (x \cdot x) - x \cdot 8 + 5 x + 5 \cdot 8}{(x - 4) (x + 8)}$

Multiply $x$ by $x$ .

$\frac{6 - x^{2} - x \cdot 8 + 5 x + 5 \cdot 8}{(x - 4) (x + 8)}$

Multiply $8$ by $- 1$ .

$\frac{6 - x^{2} - 8 x + 5 x + 5 \cdot 8}{(x - 4) (x + 8)}$

Multiply $5$ by $8$ .

$\frac{6 - x^{2} - 8 x + 5 x + 40}{(x - 4) (x + 8)}$

Add $- 8 x$ and $5 x$ .

$\frac{6 - x^{2} - 3 x + 40}{(x - 4) (x + 8)}$

Add $6$ and $40$ .

$\frac{- x^{2} - 3 x + 46}{(x - 4) (x + 8)}$

Step 5

Simplify with factoring out.

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

$\frac{- (x^{2}) - 3 x + 46}{(x - 4) (x + 8)}$

Factor $- 1$ out of $- 3 x$ .

$\frac{- (x^{2}) - (3 x) + 46}{(x - 4) (x + 8)}$

Factor $- 1$ out of $- (x^{2}) - (3 x)$ .

$\frac{- (x^{2} + 3 x) + 46}{(x - 4) (x + 8)}$

Rewrite $46$ as $- 1 (- 46)$ .

$\frac{- (x^{2} + 3 x) - 1 (- 46)}{(x - 4) (x + 8)}$

Factor $- 1$ out of $- (x^{2} + 3 x) - 1 (- 46)$ .

$\frac{- (x^{2} + 3 x - 46)}{(x - 4) (x + 8)}$

Simplify the expression.

Rewrite $- (x^{2} + 3 x - 46)$ as $- 1 (x^{2} + 3 x - 46)$ .

$\frac{- 1 (x^{2} + 3 x - 46)}{(x - 4) (x + 8)}$

Move the negative in front of the fraction.

$- \frac{x^{2} + 3 x - 46}{(x - 4) (x + 8)}$

Simplify 6/(x^2+4x-32)-(x-5)/(x-4)

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