How do you factor 32x^2y - 8y?

You can collect $8 y$ between the two terms to get:
$8 y \left(4 {x}^{2} - 1\right) = 8 y \left(2 x + 1\right) \left(2 x - 1\right)$
Using the fact that $\left({a}^{2} - {b}^{2}\right) = \left(a + b\right) \left(a - b\right)$