If $$a + b + c = 0$$, then a factor of the expression $$(a+b)^{3}+(b+c)^{3}+(c+a)^{3}$$ is

Given : $$a + b + c = 0$$

=> $$(a+b)=-c$$

Cubing both sides, => $$(a+b)^3=(-c)^3$$

=> $$a^3+b^3+3ab(a+b)=-c^3$$

=> $$a^3+b^3+3ab(-c)=-c^3$$

=> $$a^3+b^3+c^3=3abc$$ --------------(i)

Expression : $$(a+b)^{3}+(b+c)^{3}+(c+a)^{3}$$

= $$[a^3+b^3+3ab(a+b)]+[b^3+c^3+3bc(b+c)]+[c^3+a^3+3ca(c+a)]$$

= $$[a^3+b^3+3ab(-c)]+[b^3+c^3+3bc(-a)]+[c^3+a^3+3ca(-b)]$$

= $$2(a^3+b^3+c^3)-9abc$$

Substituting value from equation (i), we get :

= $$2(3abc)-9abc=-3abc$$

$$\therefore$$ $$abc$$ is a factor of the given expression.

=> Ans - (A)