(i) The binomial expansion of \((1 - 2x)^5\) is given by:

\((1 - 2x)^5 = \sum_{k=0}^{5} \binom{5}{k} (1)^{5-k} (-2x)^k\)

For \(x^4\), \(k = 4\):

Coefficient = \(\binom{5}{4} (1)^{1} (-2)^4 = 5 \times 16 = 80\)

For \(x^5\), \(k = 5\):

Coefficient = \(\binom{5}{5} (1)^{0} (-2)^5 = 1 \times (-32) = -32\)

(ii) In the expansion of \((1 + px)(1 - 2x)^5\), the term in \(x^5\) is:

\((-32) + 5p(1)(-2)^4 = 0\)

\(-32 + 80p = 0\)

\(80p = 32\)

\(p = \frac{32}{80} = \frac{2}{5}\)