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

Assume \(f(x) = \frac{A}{x-1} + \frac{B}{3x+2}\).

Multiply through by \((x-1)(3x+2)\) to clear the denominators:

\(12x^2 + 4x - 1 = A(3x+2) + B(x-1)\).

Expand and equate coefficients:

\(12x^2 + 4x - 1 = (3A + B)x + (2A - B)\).

Equating coefficients gives:

\(3A + B = 12\)

\(2A - B = 4\)

Solving these equations, we find \(A = 4\) and \(B = 3\).

Thus, \(f(x) = \frac{4}{x-1} + \frac{3}{3x+2}\).

(ii) Expansion in ascending powers of \(x\):

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

Expand \(\frac{1}{3x+2}\) as \(\frac{1}{2} - \frac{3}{4}x + \frac{9}{8}x^2 - \ldots\)

Combine these expansions:

\(f(x) = 4(-1 - x - x^2) + 3(\frac{1}{2} - \frac{3}{4}x + \frac{9}{8}x^2)\)

\(= -4 - 4x - 4x^2 + \frac{3}{2} - \frac{9}{4}x + \frac{27}{8}x^2\)

Combine like terms:

\(= \frac{1}{2} - \frac{9}{4}x - \frac{33}{8}x^2\)