TITLE: Need algebra tip about $a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$ for sides of a triangle
QUESTION [6 upvotes]: I just got a long expression:
$$a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$$
and I need to prove its less than zero for every $a$, $b$, and $c$ which are triangle sides
I really need tips how to handle such large expressions so they can be more useful for me.
I know that I should probably get something  sort of $(a+b+c)^2$ but I cant really find a way to do it , I tried some ways but a general rule that helps would be useful ...

REPLY [2 votes]: As others have mentioned,it follows in quite a straightforward  way from Heron.However,we do what you initially suggested.This method,as you may already know,is called completing the square.My hint goes as follows:
Compare your expression with the expansion of $(a^2-b^2-c^2)^2$. What can you add or subtract from your expression to get a square?Finally,complete the factorization with repeated use of the identity $a^2-b^2=(a+b)(a-b)$
MOTIVATION: As you said,it was fairly easy to realize that we have an expression resembling $(a^2+b^2+c^2)^2$, but the minus signs made it clear that it wasn't what we should be looking for.The minus signs also motivated me to compare that expression with $(a^2-b^2-c^2)^2$.The rest was quite simple.