$\frac{m + n}{m - n} - \frac{m^2 + n^2}{m^2 - n^2} = \frac{m + n}{m - n} - \frac{m^2 + n^2}{(m - n)(m + n)} = \frac{(m + n)(m + n) - (m^2 + n^2)}{(m - n)(m + n)} = \frac{(m + n)^2 - (m^2 + n^2)}{(m - n)(m + n)} = \frac{m^2 + 2mn + n^2 - m^2 - n^2}{(m - n)(m + n)} = \frac{2mn}{m^2 - n^2}$

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

$\frac{2a}{4a^2 - 1} - \frac{a + 4}{2a^2 + a} = \frac{2a}{(2a - 1)(2a + 1)} - \frac{a + 4}{a(2a + 1)} = \frac{2a * a - (2a - 1)(a + 4)}{a(2a - 1)(2a + 1)} = \frac{2a^2 - (2a^2 - a + 8a - 4)}{a(2a - 1)(2a + 1)} = \frac{2a^2 - 2a^2 + a - 8a + 4}{a(2a - 1)(2a + 1)} = \frac{4 - 7a}{a(4a^2 - 1)}$

$\frac{b - 2}{b^2 + 6b + 9} - \frac{b}{b^2 - 9} = \frac{b - 2}{(b + 3)^2} - \frac{b}{(b - 3)(b + 3)} = \frac{(b - 2)(b - 3) - b(b + 3)}{(b + 3)^2(b - 3)} = \frac{b^2 - 2b - 3b + 6 - b^2 - 3b}{(b + 3)^2(b - 3)} = \frac{6 - 8b}{(b + 3)^2(b - 3)}$

$\frac{x - 6}{x^2 + 3x} + \frac{x}{x + 3} - \frac{x - 3}{x} = \frac{x - 6}{x(x + 3)} + \frac{x}{x + 3} - \frac{x - 3}{x} = \frac{x - 6 + x * x - (x - 3)(x + 3)}{x(x + 3)} = \frac{x - 6 + x^2 - (x^2 - 9)}{x(x + 3)} = \frac{x - 6 + x^2 - x^2 + 9}{x(x + 3)} = \frac{x + 3}{x(x + 3)} = \frac{1}{x}$

$\frac{y + 2}{y - 2} - \frac{y - 2}{y + 2} - \frac{16}{y^2 - 4} = \frac{y + 2}{y - 2} - \frac{y - 2}{y + 2} - \frac{16}{(y - 2)(y + 2)} = \frac{(y + 2)(y + 2) - (y - 2)(y - 2) - 16}{(y - 2)(y + 2)} = \frac{(y + 2)^2 - (y - 2)^2 - 16}{(y - 2)(y + 2)} = \frac{y^2 + 4y + 4 - (y^2 - 4y + 4) - 16}{(y - 2)(y + 2)} = \frac{y^2 + 4y + 4 - y^2 + 4y - 4 - 16}{(y - 2)(y + 2)} = \frac{8y - 16}{(y - 2)(y + 2)} = \frac{8(y - 2)}{(y - 2)(y + 2)} = \frac{8}{y + 2}$