$\frac{b + 4}{b^2 - 6b + 9} : \frac{b^2 - 16}{2b - 6} - \frac{2}{b - 4} = \frac{b + 4}{(b - 3)^2} : \frac{(b - 4)(b + 4)}{2(b - 3)} - \frac{2}{b - 4} = \frac{b + 4}{(b - 3)^2} * \frac{2(b - 3)}{(b - 4)(b + 4)} - \frac{2}{b - 4} = \frac{1}{b - 3} * \frac{2}{b - 4} - \frac{2}{b - 4} = \frac{2}{b - 4}(\frac{1}{b - 3} - 1) = \frac{2}{b - 4} * \frac{1 - (b - 3)}{b - 3} = \frac{2}{b - 4} * \frac{1 - b + 3}{b - 3} = \frac{2}{b - 4} * \frac{4 - b}{b - 3} = \frac{2}{b - 4} * \frac{b - 4}{3 - b} = \frac{2}{1} * \frac{1}{3 - b} = \frac{2}{3 - b}$

$(\frac{m - 1}{m + 1} - \frac{m + 1}{m - 1}) : \frac{4m}{m^2 - 1} = \frac{(m - 1)^2 - (m + 1)^2}{(m - 1)(m + 1)} : \frac{4m}{m^2 - 1} = \frac{m^2 - 2m + 1 - (m^2 + 2m + 1)}{m^2 - 1} * \frac{m^2 - 1}{4m} = \frac{m^2 - 2m + 1 - m^2 - 2m - 1}{1} * \frac{1}{4m} = \frac{-4m}{4m} = -1$

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

$(\frac{2a - 3}{a^2 - 4a + 4} - \frac{a - 1}{a^2 - 2a}) : \frac{a^2 - 2}{a^3 - 4a} = (\frac{2a - 3}{(a - 2)^2} - \frac{a - 1}{a(a - 2)}) : \frac{a^2 - 2}{a(a^2 - 4)} = \frac{a(2a - 3) - (a - 1)(a - 2)}{a(a - 2)^2} : \frac{a^2 - 2}{a(a - 2)(a + 2)} = \frac{2a^2 - 3a - (a^2 - a - 2a + 2)}{a(a - 2)^2} * \frac{a(a - 2)(a + 2)}{a^2 - 2} = \frac{2a^2 - 3a - a^2 + a + 2a - 2}{a - 2} * \frac{a + 2}{a^2 - 2} = \frac{a^2 - 2}{a - 2} * \frac{a + 2}{a^2 - 2} = \frac{1}{a - 2} * \frac{a + 2}{1} = \frac{a + 2}{a - 2}$