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

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

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

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

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

$\frac{p}{3p - 1} - \frac{p}{1 + 3p} = \frac{p(3p + 1) - p(3p - 1)}{(3p - 1)(3p + 1)} = \frac{3p^2 + p - 3p^2 + p}{9p^2 - 1} = \frac{2p}{9p^2 - 1}$