Simplify m/(m^2-4)+2/(3m+6)

$\frac{m}{m^{2} - 4} + \frac{2}{3 m + 6}$

Simplify each term.

Simplify the denominator.

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

$\frac{m}{m^{2} - 2^{2}} + \frac{2}{3 m + 6}$

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

$\frac{m}{(m + 2) (m - 2)} + \frac{2}{3 m + 6}$

Factor $3$ out of $3 m + 6$ .

Factor $3$ out of $3 m$ .

$\frac{m}{(m + 2) (m - 2)} + \frac{2}{3 (m) + 6}$

Factor $3$ out of $6$ .

$\frac{m}{(m + 2) (m - 2)} + \frac{2}{3 m + 3 \cdot 2}$

Factor $3$ out of $3 m + 3 \cdot 2$ .

$\frac{m}{(m + 2) (m - 2)} + \frac{2}{3 (m + 2)}$

To write $\frac{m}{(m + 2) (m - 2)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{m}{(m + 2) (m - 2)} \cdot \frac{3}{3} + \frac{2}{3 (m + 2)}$

To write $\frac{2}{3 (m + 2)}$ as a fraction with a common denominator, multiply by $\frac{m - 2}{m - 2}$ .

$\frac{m}{(m + 2) (m - 2)} \cdot \frac{3}{3} + \frac{2}{3 (m + 2)} \cdot \frac{m - 2}{m - 2}$

Write each expression with a common denominator of $(m + 2) (m - 2) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{m}{(m + 2) (m - 2)}$ by $\frac{3}{3}$ .

$\frac{m \cdot 3}{(m + 2) (m - 2) \cdot 3} + \frac{2}{3 (m + 2)} \cdot \frac{m - 2}{m - 2}$

Multiply $\frac{2}{3 (m + 2)}$ by $\frac{m - 2}{m - 2}$ .

$\frac{m \cdot 3}{(m + 2) (m - 2) \cdot 3} + \frac{2 (m - 2)}{3 (m + 2) (m - 2)}$

Reorder the factors of $(m + 2) (m - 2) \cdot 3$ .

$\frac{m \cdot 3}{3 (m + 2) (m - 2)} + \frac{2 (m - 2)}{3 (m + 2) (m - 2)}$

Combine the numerators over the common denominator.

$\frac{m \cdot 3 + 2 (m - 2)}{3 (m + 2) (m - 2)}$

Simplify the numerator.

Move $3$ to the left of $m$ .

$\frac{3 \cdot m + 2 (m - 2)}{3 (m + 2) (m - 2)}$

Apply the distributive property.

$\frac{3 m + 2 m + 2 \cdot - 2}{3 (m + 2) (m - 2)}$

Multiply $2$ by $- 2$ .

$\frac{3 m + 2 m - 4}{3 (m + 2) (m - 2)}$

Add $3 m$ and $2 m$ .

$\frac{5 m - 4}{3 (m + 2) (m - 2)}$