$(\frac{1}{y} + \frac{2}{x - y})(x - \frac{x^2 + y^2}{x + y}) = \frac{x - y + 2y}{y(x - y)} * \frac{x(x + y) - (x^2 + y^2)}{x + y} = \frac{x + y}{y(x - y)} * \frac{x^2 + xy - x^2 - y^2}{x + y} = \frac{xy - y^2}{y(x - y)} = \frac{y(x - y)}{y(x - y)} = 1$

$(a + b - \frac{2ab}{a + b}) : (\frac{a - b}{a + b} + \frac{b}{a}) = \frac{(a + b)^2 - 2ab}{a + b} : \frac{a(a - b) + b(a + b)}{a(a + b)} = \frac{a^2 + 2ab + b^2 - 2ab}{a + b} * \frac{a(a + b)}{a^2 - ab + ab + b^2} = \frac{a^2 + b^2}{1} * \frac{a}{a^2 - b^2} = a$

$(x^2 - 1)(\frac{1}{x - 1} - \frac{1}{x + 1} + 1) = \frac{(x - 1)(x + 1)}{1} * \frac{x + 1 - (x - 1) + x^2 - 1}{(x - 1)(x + 1)} = x^2 + 1$

$m + 1 - \frac{1}{1 - m} : (m - \frac{m^2}{m - 1}) = \frac{(1 + m)(1 - m) - 1}{1 - m} : \frac{m(m - 1) - m^2}{m - 1} = \frac{m^2}{1 - m} * \frac{m - 1}{(-m)} = m$