How do you distribute this negative? So I have $-(x - 2)^2$. Do I rewrite it as $-(x - 2)\cdot-(x - 2)$ and distribute the negative to the inside making it $(-x + 2)(-x + 2)$ or add the negative at the end of doing FOIL?
The minus sign is not in the scope of the exponent, so apply the minus AFTER squaring what's in parentheses.

To indicate the other, you would have to write $(-(x-2))^2$.

In the absence of inclusion symbols such as parentheses, brackets, braces, etc., exponents only apply to the immediately preceding factor that carries them.
Just square $(x-2)$ and negate it, i.e. multiply it by $-1$