To expand \((1 + x) \sqrt{(1 - 2x)}\), we first expand \(\sqrt{(1 - 2x)}\) using the binomial series for \((1 + u)^n\) where \(n = \frac{1}{2}\) and \(u = -2x\).

The expansion is:

\(1 + \frac{1}{2}(-2x) + \frac{1}{2} \left( \frac{1}{2} - 1 \right) \frac{(-2x)^2}{2!} + \ldots\)

Simplifying, we get:

\(1 - x + \frac{1}{2}(-2x)^2 \cdot \frac{-1}{2} = 1 - x + \frac{1}{2} \cdot 2x^2 = 1 - x + x^2\)

Now, multiply this by \((1 + x)\):

\((1 + x)(1 - x + x^2) = 1 - x + x^2 + x - x^2 + x^3\)

Combine like terms:

\(1 + 0x - \frac{3}{2}x^2 + x^3\)

Thus, the expansion up to \(x^2\) is:

\(1 - \frac{3}{2}x^2\)