Explanation

(a)
$\frac{x - 4}{x^{2} - 5x + 6} = \frac{x - 4}{(x - 2)(x - 3)}$
$= \frac{2}{(x - 2)} - \frac{1}{(x - 3)}$
$= 2(x - 2)^{- 1} - (x - 3)^{- 1}$
$= 2( - 2)^{- 1}\left( 1 - \frac{x}{2} \right)^{- 1} - ( - 3)^{- 1}\left( 1 - \frac{x}{3} \right)^{- 1}$
$= - \left\lbrack 1 + \left( \frac{x}{2} \right) + \left( \frac{x}{2} \right)^{2} + \left( \frac{x}{2} \right)^{3} + ... \right\rbrack + \frac{1}{3}\left\lbrack 1 + \left( \frac{x}{3} \right) + \left( \frac{x}{3} \right)^{2} + \left( \frac{x}{3} \right)^{3} + ... \right\rbrack$
$\therefore$ Coefficient of $x^{3}$ in $\frac{x - 4}{x^{2} - 5x + 6}$
$= - \left( \frac{1}{2} \right)^{3} + \frac{1}{3}\left( \frac{1}{3} \right)^{3} = - \frac{1}{8} + \frac{1}{81} = - \frac{73}{648}$