(i) Use the binomial expansion formula: \((1 + ax)^n = 1 + n(ax) + \frac{n(n-1)}{2!}(ax)^2 + \ldots\)

For \((1 - 2x^2)^8\), the first three terms are:

\(1 + 8(-2x^2) + \binom{8}{2}(-2x^2)^2\)

\(= 1 - 16x^2 + 112x^4\)

(ii) Expand \((2 - x^2)(1 - 16x^2 + 112x^4)\).

The coefficient of \(x^4\) is found by considering:

\((2 \times 112) - (-16) = 240\)