Factorize: $a^3(b+c)+b^3(a+c)+c^3(a+b)$.
I found this question on a high school textbook but it seems impossible to be further factorized. The best I can get is:
$a^3(b+c)+b^3(a+c)+c^3(a+b) = (a+b+c)(a^3+b^3+c^3)-a^4-b^4-c^4$.
Since this question is from a high school textbook, I assume the field that we are working on is $\mathbb{R}$. We can not further factorize $a^4+b^4+c^4$ so there is no common factor with the previous term. Thus this can not be further factorized.
I am wondering if my conclusion is correct or not. I am also curious if we can factorize it on a bigger field. Please help.