Answer
ii. $\frac{A}{x-1} + \frac{Bx + C}{x^2 + 4}$
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 $\frac{2x+8}{(x-1)(x^2 + 4)}$
Factor the denominator, then split the factors into their respective fractions:
$\frac{2x+8}{(x-1)(x^2 + 4)}$ =$\frac{A}{x-1} + \frac{Bx + C}{x^2 + 4}$
(Note: Bx + C are present because the denominators are of the 2nd power, so the numerator has to be of the 1st power (1st degree, so x))
This matches with option ii presented in the problem, thus that is the answer.
