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

Assume \(\frac{5x^2 + 8x - 3}{(x-2)(2x^2 + 3)} = \frac{A}{x-2} + \frac{Bx + C}{2x^2 + 3}\).

Multiply through by the denominator: \(5x^2 + 8x - 3 = A(2x^2 + 3) + (Bx + C)(x-2)\).

Equate coefficients to find \(A = 3\), \(B = -1\), \(C = 6\).

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

Use the expansions:

\((x-2)^{-1} = \frac{1}{2} \left[ 1 - \left( \frac{x}{2} \right) + \left( \frac{x}{2} \right)^2 + \ldots \right]\)

\((2x^2 + 3)^{-1} = \frac{1}{3} \left[ 1 - \frac{2x^2}{3} + \ldots \right]\)

Combine these with the partial fractions:

\(\frac{A}{x-2} = \frac{3}{2} \left[ 1 - \frac{x}{2} + \left( \frac{x}{2} \right)^2 + \ldots \right]\)

\(\frac{Bx + C}{2x^2 + 3} = \frac{-x + 6}{3} \left[ 1 - \frac{2x^2}{3} + \ldots \right]\)

Combine and simplify to get the expansion:

\(\frac{1}{2} - \frac{13}{12}x - \frac{41}{24}x^2\).