Выполни действия: a) $(\frac{1}{y} + \frac{2}{x-y})(\frac{x^2 + y^2}{x+y})$; б) $(a + b - \frac{2ab}{a+b}):(\frac{a-b}{a+b} + \frac{b}{a})$; в) $(x^2-1)(\frac{1}{x-1} - \frac{1}{x+1} + 1)$; г) $(m + 1 - \frac{1}{1-m}):(m - \frac{m^2}{m-1})$

a) $\left(\frac{1}{y} + \frac{2}{x-y}\right)\left(x - \frac{x^2 + y^2}{x+y}\right) = \left(\frac{x-y+2y}{y(x-y)}\right)\left(\frac{x(x+y) - (x^2 + y^2)}{x+y}\right) = \frac{x+y}{y(x-y)} \cdot \frac{x^2 + xy - x^2 - y^2}{x+y} = \frac{xy - y^2}{y(x-y)} = \frac{y(x-y)}{y(x-y)} = 1$ б) $\left(a + b - \frac{2ab}{a+b}\right) : \left(\frac{a-b}{a+b} + \frac{b}{a}\right) = \frac{(a+b)^2 - 2ab}{a+b} : \frac{a(a-b) + b(a+b)}{a(a+b)} = \frac{a^2 + 2ab + b^2 - 2ab}{a+b} : \frac{a^2 - ab + ab + b^2}{a(a+b)} = \frac{a^2 + b^2}{a+b} : \frac{a^2 + b^2}{a(a+b)} = \frac{a^2 + b^2}{a+b} \cdot \frac{a(a+b)}{a^2 + b^2} = a$ в) $(x^2 - 1)\left(\frac{1}{x-1} - \frac{1}{x+1} + 1\right) = (x^2 - 1)\left(\frac{(x+1) - (x-1) + (x-1)(x+1)}{(x-1)(x+1)}\right) = (x^2 - 1)\left(\frac{x+1 - x+1 + x^2 - 1}{x^2 - 1}\right) = (x^2 - 1)\left(\frac{x^2 + 1}{x^2 - 1}\right) = x^2 + 1$ г) $\left(m + 1 - \frac{1}{1-m}\right) : \left(m - \frac{m^2}{m-1}\right) = \frac{(m+1)(1-m) - 1}{1-m} : \frac{m(m-1) - m^2}{m-1} = \frac{m - m^2 + 1 - m - 1}{1-m} : \frac{m^2 - m - m^2}{m-1} = \frac{-m^2}{1-m} : \frac{-m}{m-1} = \frac{-m^2}{1-m} \cdot \frac{m-1}{-m} = \frac{-m^2}{-(m-1)} \cdot \frac{m-1}{-m} = \frac{-m^2}{-m} = m$