First, find the coefficient of \(x^3\) in \((1 - 2x)^5\). The general term in the binomial expansion of \((1 - 2x)^5\) is given by:

\(\binom{5}{r} (1)^{5-r} (-2x)^r = \binom{5}{r} (-2)^r x^r\).

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

\(\binom{5}{3} (-2)^3 x^3 = 10(-8)x^3 = -80x^3\).

Thus, the coefficient of \(x^3\) in \((1 - 2x)^5\) is \(-80\).

Next, find the coefficient of \(x^2\) in \((1 - 2x)^5\) to combine with \(kx\) from \((1 + kx)\):

For \(x^2\), set \(r = 2\):

\(\binom{5}{2} (-2)^2 x^2 = 10(4)x^2 = 40x^2\).

Thus, the coefficient of \(x^2\) in \((1 - 2x)^5\) is \(40\).

The coefficient of \(x^3\) in \((1 + kx)(1 - 2x)^5\) is given by:

\(40k - 80 = 20\).

Solving for \(k\):

\(40k - 80 = 20\)

\(40k = 100\)

\(k = \frac{100}{40} = \frac{5}{2}\).