factrorize $-a^3+a^2b+a^2c+ab^2-2abc+ac^2-b^3+b^2c+bc^2-c^3$

Question

factrorize $-a^3+a^2b+a^2c+ab^2-2abc+ac^2-b^3+b^2c+bc^2-c^3$

or $$ -(a^3 + b^3 + c^3) + (a^2 b + a b^2 + b^2 c + b c^2 + c^2 a + c a^2) - 2(abc) $$

It is a so long polynomial I don't have any ideas for starting factoring what should I do?

Answer 1

$$ ( a +b - c) ( b + c - a ) ( c + a - b) $$ This is one factor short of Heron's formula, $$ (a + b + c) ( a +b - c) ( b + c - a ) ( c + a - b) = 2 (b^2 c^2 + c^2 a^2 + a^2 b^2) - (a^4 + b^4 + c^4) $$