50. Упростите выражение: a) (\frac{2m +1}{2m-1} - \frac{2m -1}{2m + 1}): \frac{4m}{10m - 5};

a)

$$(\frac{2m +1}{2m-1} - \frac{2m -1}{2m + 1}): \frac{4m}{10m - 5} = (\frac{(2m+1)^2 - (2m-1)^2}{(2m-1)(2m+1)}) : \frac{4m}{5(2m-1)} = \frac{(4m^2+4m+1) - (4m^2-4m+1)}{(2m-1)(2m+1)} \cdot \frac{5(2m-1)}{4m} = \frac{4m^2+4m+1 - 4m^2+4m-1}{(2m-1)(2m+1)} \cdot \frac{5(2m-1)}{4m} = \frac{8m}{(2m-1)(2m+1)} \cdot \frac{5(2m-1)}{4m} = \frac{2 \cdot 5}{2m+1} = \frac{10}{2m+1}$$.

Ответ: $$\frac{10}{2m+1}$$