$\frac{mn + 1}{m + n} + \frac{mn - 1}{m - n} = \frac{(mn + 1)(m - n) + (mn - 1)(m + n)}{(m + n)(m - n)} = \frac{m^2n + m - mn^2 - n + m^2n - m + mn^2 - n}{(m + n)(m - n)} = \frac{m^2n - n + m^2n - n}{(m + n)(m - n)} = \frac{2n(m^2 - 1)}{m^2 - n^2}$

$\frac{x + 4a}{3a + 3x} - \frac{a - 4x}{3a - 3x} = \frac{(x + 4a)(a - x) - (a - 4x)(a + x)}{3(a + x)(a - x)} = \frac{ax + 4a - x^2 - 4ax - (a^2 - 4ax + ax - 4x^2)}{3(a + x)(a - x)} = \frac{3a^2 + 3x^2}{3(a + x)(a - x)} = \frac{3(a^2 + x^2)}{3(a + x)(a - x)} = \frac{a^2 + x^2}{a^2 - x^2}$