(i) Express \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in partial fractions:

Assume \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2} = \frac{A}{1 - 2x} + \frac{B}{2 + x} + \frac{C}{(2 + x)^2}\).

Multiply through by the denominator \((1 - 2x)(2 + x)^2\) to get:

\(4 + 5x - x^2 = A(2 + x)^2 + B(1 - 2x)(2 + x) + C(1 - 2x)\).

Expand and equate coefficients to find \(A = 1\), \(B = 1\), \(C = -2\).

(ii) Expand \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in ascending powers of \(x\):

Use the partial fractions: \(\frac{1}{1 - 2x}\), \(\frac{1}{2 + x}\), and \(\frac{-2}{(2 + x)^2}\).

Expand each term using binomial series:

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

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

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

Combine the expansions up to \(x^2\):

\(1 + \frac{9}{4}x + \frac{15}{4}x^2\).