Упростите выражение: (a-b+4ab/(a-b))(4a^2/(a^2+2ab+b^2)-2a/(a+b)).

Ответ:

\[\left( a - b + \frac{4ab}{a - b} \right)\left( \frac{4a^{2}}{a^{2} + 2ab + b^{2}} - \frac{2a}{a + b} \right) =\]

\[= 2a\]

\[1)\ (a - b)^{\backslash a - b} + \frac{4ab}{a - b} =\]

\[= \frac{a^{2} - 2ab + b^{2} + 4ab}{a - b} = \frac{a^{2} + 2ab + b^{2}}{a - b} =\]

\[= \frac{(a + b)^{2}}{a - b}\ \]

\[2)\ \frac{4a^{2}}{a^{2} + 2ab + b^{2}} - \frac{2a}{a + b} = \frac{4a^{2}}{(a + b)^{2}} -\]

\[- \frac{2a^{\backslash a + b}}{a + b} = \frac{4a^{2} - 2a^{2} - 2ab}{(a + b)^{2}} = \frac{2a^{2} - 2ab}{(a + b)^{2}} =\]

\[= \frac{2a(a - b)}{(a + b)^{2}}\]

\[3)\ \ \frac{(a + b)^{2}}{a - b} \cdot \frac{2a(a - b)}{(a + b)^{2}} = 2a\]