TITLE: How to factor $2b^2c^2 + 2c^2a^2 + 2a^2b^2 -a^4-b^4-c^2$?
QUESTION [2 upvotes]: The term is: $2b^2c^2 + 2c^2a^2 + 2a^2b^2 -a^4-b^4-c^2$ 
And the answer is : $(a+b+c)(b+c-a)(c+a-b)(a+b-c)$
I have tried a lot, but could't accomplish. Please don't bring up any complex method, it is just a high school math problem. But in vain I just can't do it.

REPLY [5 votes]: $$2a^2b^2+2b^2c^2+2c^2a^2-a^4-b^4-c^4$$
$$=(2bc)^2-\{(a^2)^2+(b^2)^2+(c^2)^2-2a^2b^2+2b^2c^2-2a^2c^2\}$$
$$=(2bc)^2-(a^2-b^2-c^2)^2$$
$$=(2bc+a^2-b^2-c^2)(2bc-a^2+b^2+c^2)$$
$$=\{a^2-(b-c)^2\}\{(b+c)^2-a^2\}$$
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