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