(a) Express \(f(x)\) in the form \(\frac{A}{2+x} + \frac{B+Cx}{3+x^2}\).

Equating coefficients, solve for \(A, B,\) and \(C\):

\(A = 4, \; B = 1, \; C = -2\).

(b) Expand \(\left(1 + \frac{1}{2}x\right)^{-1}\) and \(\left(1 + \frac{1}{3}x^2\right)^{-1}\) up to \(x^2\):

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

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

Substitute back into the partial fractions and simplify:

\(2\left(1 - \frac{1}{2}x + \left(\frac{1}{2}x\right)^2\right) + \frac{1}{3}(1 - 2x)\left(1 - \frac{1}{3}x^2\right)\)

Multiply out and collect terms up to \(x^2\):

\(\frac{7}{3} - \frac{5}{3}x + \frac{7}{18}x^2\)