(a)(i) Expand \((1 + 2x)^5\) using the binomial theorem:

\((1 + 2x)^5 = 1 + 5(2x) + \frac{5 \times 4}{2}(2x)^2 = 1 + 10x + 40x^2\).

(a)(ii) Expand \((1 - ax)^6\) using the binomial theorem:

\((1 - ax)^6 = 1 - 6(ax) + \frac{6 \times 5}{2}(ax)^2 = 1 - 6ax + 15a^2x^2\).

(b) To find the coefficient of \(x^2\) in \((1 + 2x)^5(1 - ax)^6\), consider the terms that contribute to \(x^2\):

1. \(x^2\) term from \((1 + 2x)^5\) and constant term from \((1 - ax)^6\): \(40x^2 \times 1 = 40x^2\).

2. \(x\) term from \((1 + 2x)^5\) and \(x\) term from \((1 - ax)^6\): \(10x \times (-6ax) = -60ax^2\).

3. Constant term from \((1 + 2x)^5\) and \(x^2\) term from \((1 - ax)^6\): \(1 \times 15a^2x^2 = 15a^2x^2\).

Combine these to find the total coefficient of \(x^2\):

\(40 - 60a + 15a^2 = -5\).

Simplify and solve the quadratic equation:

\(15a^2 - 60a + 40 = -5\).

\(15a^2 - 60a + 45 = 0\).

Divide by 15:

\(a^2 - 4a + 3 = 0\).

Factorize:

\((a - 1)(a - 3) = 0\).

Thus, \(a = 1\) and \(a = 3\).