\[\frac{ab - 2b - 6 + 3a}{a^{2} - 4} =\]
\[= \frac{b(a - 2) + 3(a - 2)}{a^{2} - 4} =\]
\[= \frac{(a - 2)(b + 3)}{(a - 2)(a + 2)} = \frac{b + 3}{a + 2}.\]