several ways you can think about this. Here's one:

when you factor a term from an expression, you actually divide each term in the expression by the factored-out term in order to factor it out. for example:

Say you wanted to factor $\displaystyle \displaystyle x^3 + x$. Of course you would factor out the common $\displaystyle \displaystyle x$ to get $\displaystyle \displaystyle x(x^2 + 1)$

But where did the $\displaystyle \displaystyle x^2$ and the $\displaystyle \displaystyle +1$ come from? you got them by dividing each of the original terms by what you are factoring out. You factored out an $\displaystyle \displaystyle x$, so divide the $\displaystyle \displaystyle x^3$ by $\displaystyle \displaystyle x$, and that's where your $\displaystyle \displaystyle x^2$ comes from. Your 1 comes from dividing $\displaystyle \displaystyle x$ by $\displaystyle \displaystyle x$.

So in other words, what you did was $\displaystyle \displaystyle x \left( \frac {x^3}x + \frac xx \right) = x(x^2 + 1)$

(you can also think of this as multiplying and dividing by x at the same time. this way, you're multiplying by 1 and so you're not changing the value of your expression, just how it looks)

Similarly, $\displaystyle \displaystyle b - a = -1 \left( \frac b{-1} - \frac a{-1} \right) = -(a - b)$.

We usually don't think of factoring in this complicated way, but that's what we're doing when we factor something. as for factoring out -1, you just think of it as factoring out a negative sign, when you do this, you change all the signs of the terms you factored the -1 from. so b - a = -(a - b), the +b became -b and the -a became +a when you factor out a sign. If you multiply out, you get the original expression, which is a good way to check yourself.

Capice? I don't know. I feel like I'm being way too complicated with everything today... maybe I shouldn't be explaining stuff to people