I'm trying to determine the quadrature formula when the interval is [−2, 2] and the nodes are −1, 0, and 1.
However, I get stuck on this step because I don't understand how the attached image is equal to $1/2x(x-1)$
My attempt is, $(x - 1)/(-1 - 1)$ and $(x - 2)/(-1-2)$ but that is completely off.
I suppose from your setup that $x_0 = -1$, $x_1 = 0$, and $x_2=1$. Then
\begin{align} \prod_{j=1}^2 \left(\frac{x - x_j}{x_0 - x_j}\right) &= \left(\frac{x - x_1}{x_0 - x_1}\right) \left(\frac{x - x_2}{x_0 - x_2}\right)\\ &= \left(\frac{x - 0}{(-1) - 0}\right) \left(\frac{x - 1}{(-1) - 1}\right)\\ &= \left(\frac{x}{-1}\right) \left(\frac{x - 1}{-2}\right)\\ &= \frac{x (x - 1)}{(-1)(-2)}\\ &= \frac{x (x - 1)}{2}\\ \end{align}
It looks like your first attempt was to substitute $1$ for $x_1$ and $2$ for $x_2$, which, as you found out, does not work.