# Need help to factorize.

#### Farzin

##### Junior Member
Hi,
I have been stuck on factorizing this: $$\displaystyle a^2-2ab+a^2b-2b^2$$.
I thought I could solve it by making $$\displaystyle (a+b)$$ as one factor but it didn't work then I tried to add and deduct some terms which that didn't lead me to anything either.
I don't really know how what to next.
Any help would be greatly appreciated.

#### Dr.Peterson

##### Elite Member
I'm wondering if you might have copied something wrong. With three terms of degree 2 and one of degree 3, it seems like it might not factor, though I can't be sure. One small change would make it easy.

#### Farzin

##### Junior Member
I'm wondering if you might have copied something wrong. With three terms of degree 2 and one of degree 3, it seems like it might not factor, though I can't be sure. One small change would make it easy.
No everything is correct. I know it could be much easier with a small change.

#### HallsofIvy

##### Elite Member
I would be inclined to write $$\displaystyle a^2+ 2ab+ a^2b- 2b^2$$ as $$\displaystyle a^2+ 2ab+ b^2+ a^2b- 3b^2= (a+ b)^2+ a^2b- 3b^2= (a+ b)^2+ b(a^2- 3b)$$.

That's about the best you can do.

#### Dr.Peterson

##### Elite Member
That's about all I could do, too. But when I set the expression to 0 and graph it, it seems clear that it is the product of two expressions that don't look like polynomials, which is very intriguing. I haven't figured out what they are.

@Farzin, where did this come from?

#### Farzin

##### Junior Member
That's about all I could do, too. But when I set the expression to 0 and graph it, it seems clear that it is the product of two expressions that don't look like polynomials, which is very intriguing. I haven't figured out what they are.

View attachment 20735

@Farzin, where did this come from?
Thanks for your attempts, this was given to me by a friend and we both are working on it.

#### Otis

##### Elite Member
… write $$\displaystyle a^2$$ $$\displaystyle +$$ $$\displaystyle 2ab+ a^2b- 2b^2$$ …

$$\displaystyle (a+ b)^2+ b(a^2- 3b)$$
The term 2ab should be subtracted, making the result

$$\displaystyle (a - b)^2 \,+\, b\hspace{.09em}(a^2 - 3b)$$ …

$$\;$$