Factor \(ab^3 - a^3 b + bc^3 - b^3 c + ca^3 - c^3 a\)

My calculator says

a*b^3 - a^3*b + b*c^3 - b^3*c + c*a^3 - c^3*a = (a - b)(a + c)(b + c)(a + b - c)

I'm more concerned about how you actually get there. I need to write a proof on how to factor it.

I would be interested to see someones approach as well

Can someone please help me with this? I need it by the end of tonight.