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

$\frac{x + 2}{3x + 9} - \frac{4 - x}{5x + 15} = \frac{x + 2}{3(x + 3)} - \frac{4 - x}{5(x + 3)} = \frac{5(x + 2) - 3(4 - x)}{15(x + 3)} = \frac{5x + 10 - 12 + 3x}{15(x + 3)} = \frac{8x - 2}{15(x + 3)}$

$\frac{m + 1}{m - 3} - \frac{m + 2}{m + 3} = \frac{(m + 1)(m + 3) - (m + 2)(m - 3)}{(m - 3)(m + 3)} = \frac{m^2 + m + 3m + 3 - (m^2 + 2m - 3m - 6)}{m^2 - 9} = \frac{m^2 + 4m + 3 - m^2 - 2m + 3m + 6}{m^2 - 9} = \frac{5m + 9}{m^2 - 9}$

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

$\frac{m}{3m - 2n} - \frac{3m^2 - 3mn}{9m^2 - 12m + 4n^2} = \frac{m}{3m - 2n} - \frac{3m^2 - 3mn}{(3m - 2n)^2} = \frac{m(3m - 2n) - (3m^2 - 3mn)}{(3m - 2n)^2} = \frac{3m^2 - 2mn - 3m^2 + 3mn}{(3m - 2n)^2} = \frac{mn}{(3m - 2n)^2}$

$\frac{a + 3}{a^2 - 2a} - \frac{a - 2}{5a - 10} + \frac{a + 2}{5a} = \frac{a + 3}{a(a - 2)} - \frac{a - 2}{5(a - 2)} + \frac{a + 2}{5a} = \frac{5(a + 3) - a(a - 2) + (a - 2)(a + 2)}{5a(a - 2)} = \frac{5a + 15 - a^2 + 2a + a^2 - 4}{5a(a - 2)} = \frac{7a + 11}{5a(a - 2)}$

$\frac{3}{3a - 3} - \frac{a - 1}{2a^2 - 4a + 2} = \frac{3}{3(a - 1)} - \frac{a - 1}{2(a^2 - 2a + 1)} = \frac{3}{3(a - 1)} - \frac{a - 1}{2(a - 1)^2} = \frac{3 * 2(a - 1) - 3(a - 1)}{6(a - 1)^2} = \frac{(a - 1)(6 - 3)}{6(a - 1)^2} = \frac{3}{6(a - 1)} = \frac{1}{2(a - 1)}$

$2 - \frac{14}{m - 2} - m = \frac{2(m - 2) - 14 - m(m - 2)}{m - 2} = \frac{2m - 4 - 14 - m^2 + 2m}{m - 2} = \frac{-m^2 + 4m - 18}{m - 2}$

$\frac{2x + 1}{x^2 - 6x + 9} - \frac{8}{x^2 - 9} - \frac{2x - 1}{x^2 + 6x + 9} = \frac{2x + 1}{(x - 3)^2} - \frac{8}{(x - 3)(x + 3)} - \frac{2x - 1}{(x + 3)^2} = \frac{(2x + 1)(x + 3)^2 - 8(x - 3)(x + 3) - (2x - 1)(x - 3)^2}{(x - 3)^2(x + 3)^2} = \frac{(2x + 1)(x^2 + 6x + 9) - 8(x^2 - 9) - (2x - 1)(x^2 - 6x + 9)}{(x - 3)^2(x + 3)^2} = \frac{2x^3 + 12x^2 + 18x + x^2 + 6x + 9 - 8x^2 + 72 - (2x^3 - 12x^2 + 18x - x^2 + 6x - 9)}{(x - 3)^2(x + 3)^2} = \frac{18x^2 + 90}{(x - 3)^2(x + 3)^2}$