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

. Of course you would factor out the common

to get

But where did the

and the

come from? you got them by dividing each of the original terms by what you are factoring out. You factored out an

, so divide the

by

, and that's where your

comes from. Your 1 comes from dividing

by

.

So in other words, what you did was

(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,

.

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