
I am really stumped as to how it simplifies. I should know this, but it has been a very, very, long summer.

I am really stumped as to how it simplifies. I should know this, but it has been a very, very, long summer.
Pull out a factor of $2$ and cancel:
$$ \begin{align*} \frac{4-2\sqrt{2}}{4} &= \frac{2(2 - \sqrt{2})}{2 \cdot 2}\\ &= \frac{2}{2}\cdot \frac{2-\sqrt{2}}{2}\\ &= 1 \cdot \frac{2-\sqrt{2}}{2}\\ &= \frac{2-\sqrt{2}}{2} \end{align*} $$
Notice we didn't do anything special with the $\sqrt{2}$; it may as well have just been an $x$.