Problem: Factor $$\sum_{sym}a^4b^2-a^4bc-a^3b^3+2a^3b^2c-a^2b^2c^2$$
I'm not too sure how to factor this, nor do I understand what the "sym" means at the bottom of the sigma notation. I'm guessing that this means some sort of symmetry, or cycle through, but I'm not exactly $100\%$ sure.
Any help would be okay. Including hints!
Expanding out, we (unfortunately) get $$a^4b^2-2a^4bc+a^4c^2-2a^3b^3+2a^3b^2c+2a^3bc^2-2a^3c^3+a^2b^4+2a^2b^3c-6a^2b^2c^2+2a^2bc^3+a^2c^4-2ab^4c+2ab^3c^2+2ab^2c^3-2abc^4+b^4c^2-2b^3c^3+b^2c^4\tag1$$ And factoring out $a^4$ from the first three constants, $a^3$ from the next $4$ constants, and so on, we obtain $$a^4(b^2-2bc+c^2)+a^3(-2b^3+2b^2c+2bc^2-2c^3)+a^2(b^4+2b^3c-6b^2c^2+2bc^3+c^4)+a(-2b^4c+2b^3c^2+2b^2c^3-2bc^4)+(b^4c^2-2b^3c^3+b^2c^4)\tag2$$ And factoring the expressions inside the parenthesis, we get the factorization as $$(a-b)^2(a-c)^2(b-c)^2$$ as our answer.