a^2(b^3-c^3)+b^2(c^3-a^3)+c^2(a^3-b^3)
=a^2b^3-a^2c^3+b^2c^3-a^3b^2+c^2(a^3-b^3)
=a^2b^2(b-a)+c^3(b-a)(b+a)-c^2(b-a)(b^2+ab+a^2)
=(b-a)(a^2b62+c^3b+c^3a-c^2b^2-abc^2-a^2c^2)
=(b-a)(a^2(b+c)(b-c)-c^2b(b-c)-ac^2(b-c))
=(b-a)(b-c)(a^2b+a^2c-c^2b-ac^2)
=(b-a)(b-c)(b(a-c)(a+c)+ac(a-c))
=(a-b)(b-c)(c-a)(ab+bc+ac).

.(x+y)^5-x^5-y^5
=(x+y)^5-(x^5+y^5)
=(x+y)^5-(x+y)(x^4-x^3y+x^2y^2-xy3+y^4)
=(x+y)((x^2+2xy+y^2)^2-(x^4-x^3y+x^2y^2-xy3+y^4))
=(x+y)(x^4+4x^2y^2+y^4+4x^3y+2x^2y^2+4xy^3-x^4+x^3y-x^2y^2+xy3-y^4)
=(x+y)(5x^2y^2+5x^3y+5xy^3)
=5xy(x+y)(x^2+xy+y^2).

.(y+z)(z+x)(x+y)+xyz
=(yz+xy+z^2+xz)(x+y)+xyz
=xyz+y^2z+x^2y+xy^2+xz^2+yz^2+x^2z+xyz+xyz
=y^2z+yz^2+xyz+x^2y+xy^2+xyz+x^2z+xz^2+xyz
=yz(y+z+x)+xy(x+y+z)+xz(x+z+y)
=(x+y+z)(yz+xy+xz).