a^3+b^3-c^3+3abc=?

x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

Then, a³ + b³ - c³ + 3abc
= a³ + b³ + (-c)³ - 3ab(-c)
= [a + b + (-c)] [a² + b² + (-c)² - ab - b(-c) - (-c)a]
= (a + b - c)(a² + b² + c² - ab + bc + ca)

If you put c=a+b and simplify the result is 0.
Therefore (a+b-c) is a factor.

uhh. ok.