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

Assume \(\frac{16 - 17x}{(2 + x)(3 - x)^2} = \frac{A}{2 + x} + \frac{B}{3 - x} + \frac{C}{(3 - x)^2}\).

Multiply through by the denominator \((2 + x)(3 - x)^2\) to clear the fractions:

\(16 - 17x = A(3 - x)^2 + B(2 + x)(3 - x) + C(2 + x)\).

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

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

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

Use the partial fraction decomposition:

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

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

\(\frac{-1}{(3 - x)^2} = -\frac{1}{9} - \frac{2x}{27} - \frac{3x^2}{81} + \cdots\)

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

\(f(x) = \left(1 - \frac{x}{2} + \frac{x^2}{4}\right) + \left(-\frac{2}{3} - \frac{2x}{9} - \frac{2x^2}{27}\right) + \left(-\frac{1}{9} - \frac{2x}{27} - \frac{3x^2}{81}\right)\).

Combine like terms:

\(f(x) = \frac{8}{9} - \frac{43}{54}x + \frac{7}{108}x^2\).