Given

$\rm \frac{a - b}{a + b} - \frac{a + b}{a - b} + \frac{2ab}{a^2 - b^2}$

$\rm = \frac{a - b}{a + b} \cdot \frac{a - b}{a - b} - \frac{a + b}{a - b} \cdot \frac{a + b}{a + b} + \frac{2ab}{a^2 - b^2}$

$\rm = \frac{ (a-b)(a-b)}{(a+b)(a-b)} - \frac{(a+b)(a+b)}{(a-b)(a+b)} + \frac{2ab}{a^2 - b^2}$

Using the formula for $\rm (a - b)(a + b) = a^2 - b^2$, we get,

$\rm = \frac{ (a - b)^2}{ a^2 - b^2} - \frac{ (a + b)^2}{a^2 - b^2} + \frac{2ab}{a^2 - b^2}$

$\rm = \frac{ (a - b)^2 - (a + b)^2 + 2ab}{a^2 - b^2}$

$\rm = \frac{ a^2 - 2ab + b^2 - (a^2 + 2ab + b^2) + 2ab}{a^2 - b^2}$

$\rm = \frac{ a^2 + b^2 - a^2 - 2ab - b^2}{ a^2 - b^2}$

$\rm = - \frac{ 2ab}{a^2 - b^2}$

Hence, $\rm \frac{a - b}{a + b} - \frac{a + b}{a - b} + \frac{2ab}{a^2 - b^2} = - \frac{ 2ab}{a^2 - b^2}$