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

Assume \(f(x) = \frac{A}{2-x} + \frac{Bx+C}{x^2+1}\).

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

\(A = 4, \ B = 4, \ C = 1\).

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

(ii) Show the expansion:

Use the binomial expansion for \((1-x)^{-1}\) and \((1+x^2)^{-1}\):

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

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

Multiply the expansions: \((4x+1)(1-x^2)^{-1}\) and simplify.

Combine terms to obtain: \(f(x) = 3 + 5x - \frac{1}{2}x^2 - \frac{15}{4}x^3\).