Explanation

(b)
$\frac{3x^{3} - 8x^{2} + 10}{(x - 1)^{4}} = \frac{A}{(x - 1)} + \frac{B}{(x - 1)^{2}} + \frac{C}{(x - 1)^{3}} + \frac{D}{(x - 1)^{4}}$
$\Rightarrow \ \ \ 3x^{3} - 8x^{2} + 10 = A(x - 1)^{3} + B(x - 1)^{2} + C(x - 1) + D$
Equating coefficient of different powers of $x,\ 3 = A$
$- 8 = - 3A + B\ \ \ \ \Rightarrow \ \ \ \ B = 1$
$0 = 3A - 2B + C\ \ \ \Rightarrow \ \ C = - 7$
$10 = - A + B - C + D\ \ \ \Rightarrow \ \ \ D = 5$
$\therefore$ Given expression
$= \frac{3}{x - 1} + \frac{1}{(x - 1)^{2}} - \frac{7}{(x - 1)^{3}} + \frac{5}{(x - 1)^{4}}$