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

$\frac{18}{b^2 + 3b} - \frac{6}{b} = \frac{18}{b(b + 3)} - \frac{6}{b} = \frac{18 - 6(b + 3)}{b(b + 3)} = \frac{18 - 6b - 18}{b(b + 3)} = \frac{-6b}{b(b + 3)} = -\frac{6}{b + 3}$

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

$\frac{d - 1}{2d - 8} + \frac{d}{d - 4} = \frac{d - 1}{2(d - 4)} + \frac{d}{d - 4} = \frac{d - 1 + 2d}{2(d - 4)} = \frac{3d - 1}{2(d - 4)}$

$\frac{m + 1}{3m - 15} - \frac{m - 1}{2m - 10} = \frac{m + 1}{3(m - 5)} - \frac{m - 1}{2(m - 5)} = \frac{2(m + 1) - 3(m - 1)}{6(m - 5)} = \frac{2m + 2 - 3m + 3}{6(m - 5)} = \frac{5 - m}{6(m - 5)} = -\frac{m - 5}{6(m - 5)} = -\frac{1}{6}$

$\frac{m - 2n}{6m + 6n} - \frac{m - 3n}{4m + 4n} = \frac{m - 2n}{6(m + n)} - \frac{m - 3n}{4(m + n)} = \frac{2(m - 2n) - 3(m - 3n)}{12(m + n)} = \frac{2m - 4n - 3m + 9n}{12(m + n)} = \frac{5n - m}{12(m + n)}$

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

$\frac{3x - 4y}{x^2 - 2xy} - \frac{3y - x}{xy - 2y^2} = \frac{3x - 4y}{x(x - 2y)} - \frac{3y - x}{y(x - 2y)} = \frac{y(3x - 4y) - x(3y - x)}{xy(x - 2y)} = \frac{3xy - 4y^2 - 3xy + x^2}{xy(x - 2y)} = \frac{x^2 - 4y^2}{xy(x - 2y)} = \frac{(x - 2y)(x + 2y)}{xy(x - 2y)} = \frac{x + 2y}{xy}$