(i) To find the coefficient of \(x^3\) in \((1 + 2x)^6\), use the binomial theorem:

The general term is \(\binom{6}{r} (1)^{6-r} (2x)^r\).

For \(x^3\), set \(r = 3\):

\(\binom{6}{3} (2x)^3 = 20 \times 8x^3 = 160x^3\).

Thus, the coefficient is 160.

(ii) For \((1 - 3x)(1 + 2x)^6\), expand \((1 + 2x)^6\) and find the coefficient of \(x^2\):

The term in \(x^2\) is \(\binom{6}{2} (2x)^2 = 15 \times 4x^2 = 60x^2\).

Now, consider the product \((1 - 3x)(1 + 2x)^6\):

The coefficient of \(x^3\) is given by:

\(1 \times 160 - 3 \times 60 = 160 - 180 = -20\).

Thus, the coefficient is -20.