First, expand \((1 + ax)^6\) using the binomial theorem:

The coefficient of \(x\) in \((1 + ax)^6\) is \(6ax\).

The coefficient of \(x^2\) in \((1 + ax)^6\) is \(15a^2x^2\).

Now, consider the expansion of \((1 - 2x)^2 = 1 - 4x + 4x^2\).

Multiply by \((1 + ax)^6\):

For the \(x\) term: \(6a - 4 = -1\), solving gives \(a = \frac{1}{2}\).

For the \(x^2\) term: \(15a^2 - 24a + 4 = b\), substituting \(a = \frac{1}{2}\) gives \(b = -1\).