Since this is a cyclic polynomial therefore factors are also cyclic : $f(a) = a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)$ $f(b) = b(b^2-c^2)+b(c^2-b^2)+c(b^2-b^2) = 0 \Rightarrow a-b$ is a factor of the given expression. therefore other factors are $(b-c)$ and $(c-a)$
The given expression may have a coefficient a constant factor which is non zero . Let it be $m$. $\therefore a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2) = m(a-b)(b-c)(c-a)$
Please guide further how to find this coefficient.