مخرج مشترک می‌گیریم:

$\begin{gathered}
  \left( {\frac{{a + b}}{{2a - 2b}} - \frac{{a - b}}{{2a + 2b}} + \frac{{2{b^2}}}{{{a^2} - {b^2}}}} \right) \times \frac{{{{(a - b)}^2}}}{{2b}} = \frac{{\mathop {\cancel{{2b}}}\limits^1 }}{{\mathop {\cancel{{a - b}}}\limits_1 }} \times \frac{{\mathop {\cancel{{{{(a - b)}^2}}}}\limits^{a - b} }}{{\mathop {\cancel{{2b}}}\limits_1 }} = a - b \hfill \\
  \frac{{(a + b) \times (a + b) - (a - b)(a - b)}}{{2(a - b)(a + b)}} = \frac{{{{(a + b)}^2} - {{(a - b)}^2}}}{{2({a^2} - {b^2})}} = \frac{{\mathop {\cancel{4}}\limits^2 ab}}{{\mathop {\cancel{2}}\limits_1 ({a^2} - {b^2})}} = \frac{{2ab}}{{({a^2} - {b^2})}} \hfill \\ 
\end{gathered} $

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

نکته: ${(a + b)^2} - {(a - b)^2} = 4ab$