Answer
iii. $\frac{A}{x} + \frac{B}{x-2} + \frac{C}{(x-2)^2}$
Work Step by Step
The question ONLY asks for the form of the partial fraction decomposition of the function. It says to not determine the numerical values of the coefficients.
The breakdown of a typical partial fraction is as follows:
$\frac{x}{x_1 * x_2 * x_3 * ... } = \frac{A}{x_1} + \frac{B}{x_2} + \frac{C}{x_3} + ... $
Given $\frac{4}{ x(x-2)^2}$
Factor the denominator, then split the factors into their respective fractions:
$\frac{4}{x(x-2)^2}$ =$\frac{A}{x} + \frac{B}{x-2} + \frac{C}{(x-2)^2}$
This matches with option iii presented in the problem, thus that is the answer.
