Homework help?

x^2 - y^2 + x - y

x (x + 1) - y (y + 1)

(x - y)(x + y + 1).

A big part of math is pattern recognition. A lot of homework is just fighting with stuff so you will remember the pattern when you see it again. The most common pattern is (x + a) * (x + b) = x^2 + (a + b)x + ab and the special case (x + a) * (x - a) = x^2 - a^2. When you spot the pattern you can just write the answer from memory.

x^2 - y^2 + x - y

So you stare at this, looking for a pattern. You rewrite it to see if something jumps out.

x^2 + x - y^2 - y

Then you try

x(x + 1) - y(y + 1)

And then you break that down to

(x - y)(x + y + 1)

That's all I see, but then I haven't used this stuff in sixty years. Phibonacci's answer is more complete.

Now study this page until it seems obvious: en.wikipedia.org/wiki/Polynomial_remainder_theorem