What skill do I lack to factor multivariate polynomials? Ok so I can factor easily regular quadratic polynomials, i.e. $5x^2+7x+9$ (I'm not sure whether that's prime, just made it up), and I was working on solving $y^2+(x^2+2x−2)y+(x^3−x^2−2x)$ by distributing the $y$ after the first parentheses and now I'm stuck. Am I just missing some skill in factoring multivariate polynomials? Don't tell me stuff about rings and fields because that won't help my situation. Thanks

In this case, it's going to be taking advantage of what you know, to bootstrap yourself into new territory. You basically have
$$y^2 + By + C,$$
where $B = x^2 + 2x - 2$ and $C = x^3 - x^2 - 2x\ $ don't depend on $y$; it looks a little bit like a quadratic in $y$.
Using what we know about quadratics, we "know" it should factor as $(y + p)(y + q) = y^2 + (p + q)y + pq$, if it can be factored. Now, since you know how to factor quadratics with a single variable, that means you know we're looking for a pair of things (it would be a pair of numbers, if we had a good old quadratic with one variable), call them $p$ and $q$, such that $pq = C$ and $p + q = B$.
So, let's see how would could find solutions $p, q$ to $pq = C = x^3 - x^2 - 2x$.
We can factor $$C = x^3 - x^2 - 2x = x(x^2 - x - 2) = x(x - 2)(x + 1).$$ 
More specifically since we need a pair, let's see what two polynomials multiply to $C = x(x - 2)(x + 1)$:
\begin{align*} C &= x(x^2 - x - 2) \\
&=(-x)(-x^2 + x + 2)\\
&= (x^2 - 2x)(x + 1)\\
&= (-x^2 + 2x)(-x - 1)\\
&= (x^2 + x)(x - 2)\\
&= (-x^2 - x)(-x + 2)
\end{align*}
So, we have a few possibilities for $p$ and $q$, if we only require $pq = C$. Now let's see if any of them add up to $B = x^2 + 2x - 2$.
Lo and behold, $p = x^2 + x$ and $q = x - 2$ will work. We already know they multiply to $C$, and their sum is indeed $x^2 + 2x - 2 = B$.
Thus, your polynomial factors as 
$$(y + p)(y + q) = (y + x^2 + x)(y + x - 2).$$
