(i) Express \(f(x)\) in partial fractions:

Assume \(\frac{7x^2 - 15x + 8}{(1 - 2x)(2 - x)^2} = \frac{A}{1 - 2x} + \frac{B}{2 - x} + \frac{C}{(2 - x)^2}\).

Equating coefficients, solve:

\(7 = A + 2B\)

\(-15 = -4A - 5B - 2C\)

\(8 = 4A + 2B + C\)

Solving gives \(A = 1\), \(B = 3\), \(C = -2\).

(ii) Expand \(f(x)\) in ascending powers of \(x\):

Expand each term:

\(\frac{1}{1 - 2x} = 1 + 2x + 4x^2\)

\(\frac{1}{2 - x} = \frac{1}{2}(1 + \frac{x}{2} + \frac{x^2}{4})\)

\(\frac{1}{(2 - x)^2} = \frac{1}{4}(1 + x + \frac{3x^2}{2})\)

Combine terms:

\(f(x) = \frac{1}{1 - 2x} + 3 \cdot \frac{1}{2 - x} - 2 \cdot \frac{1}{(2 - x)^2}\)

\(= (1 + 2x + 4x^2) + 3(\frac{1}{2} + \frac{x}{4} + \frac{x^2}{8}) - 2(\frac{1}{4} + \frac{x}{2} + \frac{3x^2}{8})\)

Combine and simplify:

\(= \frac{9}{4} + x + 4x^2\)