a^2b-a^2c+b^2c-b^2a+c^2a-c^2b
=a^2(b-c)+bc(b-c)-a(b^2-c^2)
=(a^2+bc)(b-c)-a(b+c)(b-c)
=(a^2+bc-ab-ac)(b-c)
=(a(a-b)-c(a-b))(b-c)
=(a-c)(a-b)(b-c)