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

Assume \(\frac{x^2 - 8x + 9}{(1-x)(2-x)^2} = \frac{A}{1-x} + \frac{B}{2-x} + \frac{C}{(2-x)^2}\).

Multiply through by \((1-x)(2-x)^2\) to clear the denominators:

\(x^2 - 8x + 9 = A(2-x)^2 + B(1-x)(2-x) + C(1-x)\).

Expand and equate coefficients to solve for \(A, B, C\):

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

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

Use the partial fractions: \(\frac{2}{1-x} - \frac{1}{2-x} + \frac{3}{(2-x)^2}\).

Expand each term using binomial series:

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

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

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

Add the expansions up to \(x^2\):

\(\frac{9}{4} + \frac{5}{2}x + \frac{39}{16}x^2\).