$\frac{4}{15x^2y^2} = \frac{4 * 2x}{15x^2y^2 * 2x} = \frac{8x}{30x^3y^2}$
$\frac{1}{10x^3y} = \frac{1 * 3y}{10x^3y * 3y} = \frac{3y}{30x^3y^2}$

$\frac{c}{6a^4b^5} = \frac{c * 3}{6a^4b^5 * 3} = \frac{3c}{18a^4b^5}$
$\frac{d}{9ab^2} = \frac{d * 2a^3b^3}{9ab^2 * 2a^3b^3} = \frac{2a^3b^3d}{18a^4b^5}$

$\frac{x}{y - 5} = \frac{x(y + 5)}{(y - 5)(y + 5)} = \frac{x(y + 5)}{y^2 - 25}$
$\frac{z}{y^2 - 25}$

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

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

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

$\frac{1 + c^2}{c^2 - 16} = \frac{1 + c^2}{(c - 4)(c + 4)} = \frac{1 + c^2}{c^2 - 16}$
$\frac{c}{4 - c} = -\frac{c}{c - 4} = -\frac{c(c + 4)}{(c - 4)(c + 4)} = -\frac{c(c + 4)}{c^2 - 16}$

$\frac{2m + 9}{m^2 + 5m + 25} = \frac{(m - 5)(2m + 9)}{(m - 5)(m^2 + 5m + 25)} = \frac{(m - 5)(2m + 9)}{m^3 - 125}$
$\frac{m}{m - 5} = \frac{m(m^2 + 5m + 25)}{(m - 5)(m^2 + 5m + 25)} = \frac{m(m^2 + 5m + 25)}{m^3 - 125}$