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

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

Simplify the denominator.

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

$\frac{6}{x^{2} - 2^{2}} + \frac{x}{x + 2}$

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 $b = 2$ .

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

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

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

Simplify terms.

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Move $- 2$ to the left of $x$ .

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

Reorder terms.

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