$\frac{5}{y - 3} + \frac{1}{y + 3} - \frac{4y - 18}{y^2 - 9} = \frac{5(y + 3) + (y - 3) - (4y - 18)}{(y - 3)(y + 3)} = \frac{5y + 15 + y - 3 - 4y + 18}{(y - 3)(y + 3)} = \frac{30}{y^2 - 9}$

$\frac{2a}{2a + 3} + \frac{5}{3 - 2a} - \frac{4a^2 + 9}{4a^2 - 9} = \frac{2a}{2a + 3} - \frac{5}{2a - 3} - \frac{4a^2 + 9}{(2a + 3)(2a - 3)} = \frac{2a(2a - 3) - 5(2a + 3) - (4a^2 + 9)}{(2a + 3)(2a - 3)} = \frac{4a^2 - 6a - 10a - 15 - 4a^2 - 9}{(2a + 3)(2a - 3)} = \frac{-16a - 24}{(2a + 3)(2a - 3)} = \frac{-8(2a + 3)}{(2a + 3)(2a - 3)} = -\frac{8}{2a - 3} = \frac{8}{3 - 2a}$

$\frac{4m}{4m^2 - 1} - \frac{2m + 1}{6m - 3} + \frac{2m - 1}{4m + 2} = \frac{4m}{(2m - 1)(2m + 1)} - \frac{2m + 1}{3(2m - 1)} + \frac{2m - 1}{2(m + 1)} = \frac{4m * 6 - 2(2m + 1)^2 + 3(2m - 1)^2}{6(2m - 1)(2m + 1)} = \frac{24m - 2(4m^2 + 4m + 1) + 3(4m^2 - 4m + 1)}{6(2m - 1)(2m + 1)} = \frac{24m - 8m^2 - 8m - 2 + 12m^2 - 12m + 3}{6(2m - 1)(2m + 1)} = \frac{4m^2 + 4m + 1}{6(2m - 1)(2m + 1)} = \frac{(2m + 1)^2}{6(2m - 1)(2m + 1)} = \frac{2m + 1}{6(2m - 1)}$

$\frac{1}{(x + y)^2} - \frac{2}{x^2 - y^2} + \frac{1}{(x - y)^2} = \frac{(x - y)^2 - 2(x^2 - y^2) + (x + y)^2}{(x + y)^2(x - y)^2} = \frac{x^2 - 2xy + y^2 - 2x^2 + 2y^2 + x^2 + 2xy + y^2}{(x + y)^2(x - y)^2} = \frac{4y^2}{(x - y)^2}$

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

$\frac{x - y}{x^2 + xy + y^2} - \frac{3xy}{x^3 - y^3} + \frac{1}{x - y} = \frac{(x - y)^2 - 3xy + x^2 + xy + y^2}{(x - y)(x^2 + xy + y^2)} = \frac{x^2 - 2xy + y^2 - 2xy + x^2 + y^2}{(x - y)(x^2 + xy + y^2)} = \frac{2(x^2 - 2xy + y^2)}{(x - y)(x^2 + xy + y^2)} = \frac{2(x - y)^2}{(x - y)(x^2 + xy + y^2)} = \frac{2(x - y)}{x^2 + xy + y^2}$