Simplify (x+1)/(x^2-16)-3/(x+4)

$\frac{x + 1}{x^{2} - 16} - \frac{3}{x + 4}$

Step 1

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

$\frac{x + 1}{x^{2} - 4^{2}} - \frac{3}{x + 4}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$\frac{x + 1}{(x + 4) (x - 4)} - \frac{3}{x + 4}$

Step 2

To write $- \frac{3}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{x + 1}{(x + 4) (x - 4)} - \frac{3}{x + 4} \cdot \frac{x - 4}{x - 4}$

Step 3

Simplify terms.

Multiply $\frac{3}{x + 4}$ by $\frac{x - 4}{x - 4}$ .

$\frac{x + 1}{(x + 4) (x - 4)} - \frac{3 (x - 4)}{(x + 4) (x - 4)}$

Combine the numerators over the common denominator.

$\frac{x + 1 - 3 (x - 4)}{(x + 4) (x - 4)}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{x + 1 - 3 x - 3 \cdot - 4}{(x + 4) (x - 4)}$

Multiply $- 3$ by $- 4$ .

$\frac{x + 1 - 3 x + 12}{(x + 4) (x - 4)}$

Subtract $3 x$ from $x$ .

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

Add $1$ and $12$ .

$\frac{- 2 x + 13}{(x + 4) (x - 4)}$

Step 5

Simplify with factoring out.

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

$\frac{- (2 x) + 13}{(x + 4) (x - 4)}$

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{- (2 x) - 1 (- 13)}{(x + 4) (x - 4)}$

Factor $- 1$ out of $- (2 x) - 1 (- 13)$ .

$\frac{- (2 x - 13)}{(x + 4) (x - 4)}$

Simplify the expression.

Rewrite $- (2 x - 13)$ as $- 1 (2 x - 13)$ .

$\frac{- 1 (2 x - 13)}{(x + 4) (x - 4)}$

Move the negative in front of the fraction.

$- \frac{2 x - 13}{(x + 4) (x - 4)}$