$\frac{2a^2 + 2}{a^2 - 1} - \frac{a + 1}{a - 1} + \frac{3a - 3}{2a + 2} = \frac{2a^2 + 2}{(a - 1)(a + 1)} - \frac{a + 1}{a - 1} + \frac{3(a - 1)}{2(a + 1)} = \frac{2(2a^2 + 2) - 2(a + 1)^2 + 3(a - 1)^2}{2(a - 1)(a + 1)} = \frac{4a^2 + 4 - 2(a^2 + 2a + 1) + 3(a^2 - 2a + 1)}{2(a - 1)(a + 1)} = \frac{4a^2 + 4 - 2a^2 - 4a - 2 + 3a^2 - 6a + 3}{2(a - 1)(a + 1)} = \frac{5a^2 - 10a + 5}{2(a - 1)(a + 1)} = \frac{5(a^2 - 2a + 1)}{2(a - 1)(a + 1)} = \frac{5(a - 1)^2}{2(a - 1)(a + 1)} = \frac{5(a - 1)}{2(a + 1)}$