Can You Factorize This Complex Algebraic Expression?

[anemone]

Here is this week's POTW:

-----

Factorize $a(b^2+c^2-a^2 )+ b(c^2+a^2-b^2 )+ c(a^2+b^2-c^2 )-2abc$.

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to the following members for their correct solution:

1. MarkFL
2. kaliprasad
3. greg1313
4. Euge

Solution from Euge:

Spoiler

We have
$$a(b^2 + c^2 - a^2) + b(c^2 + a^2 - b^2) + c(a^2 + b^2 - c^2) - 2abc$$
$$=[a(b^2 + c^2 - a^2) + 2abc] + [b(c^2 + a^2 - b^2) - 2abc] + [c(a^2 + b^2 - c^2) - 2abc]$$
$$= a[(b + c)^2 - a^2] + b[(c - a)^2 - b^2] + c[(a - b)^2 - c^2]$$
$$= a(b + c - a)(b + c + a) + b(c - a - b)(c - a + b) - c(c + b - a)(a - b + c)$$
$$= (b + c - a)[a(b + c + a) + b(c - a - b) - c(a - b + c)]$$
$$= (b + c - a)[a^2 - (b^2 - 2bc + c^2)]$$
$$= (b + c - a)[a^2 - (b - c)^2]$$
$$= (b + c - a)(a - b + c)(a + b - c).$$