Simplify 5/(x+4)-(4x)/(x^2-16)

$\frac{5}{x + 4} - \frac{4 x}{x^{2} - 16}$

Step 1

Simplify the denominator.

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

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

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 $b = 4$ .

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

Step 2

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

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

Step 3

Simplify terms.

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

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Apply the distributive property.

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

Multiply $5$ by $- 4$ .

$\frac{5 x - 20 - 4 x}{(x + 4) (x - 4)}$

Subtract $4 x$ from $5 x$ .

$\frac{x - 20}{(x + 4) (x - 4)}$