Factorize $a^4(b^2-c^2)+b^4(c^2-a^2)+c^4(a^2-b^2)$ is

$\begin{array}{l}f\left(a\right)=a^4(b^2-c^2)+b^4(c^2-a^2)+c^4(a^2-b^2)\\ f\left(b\right)=a^4(b^2-c^2)+b^4(c^2-a^2)+c^4\left(0\right)\end{array}$

$\Rightarrow b^6-b^4{c}^2+b^4{c}^2-b^6=0$

$\therefore (a-b)$is a factor

Similarly $(b-c),(c-a)$ and also $f(-b)=0$

$\Rightarrow (a+b)$ is a factor

$\Rightarrow (b+c)(c+a)$ are also factors of $f\left(a\right)$

Factors of given expression is $(a^2-b^2)(b^2-c^2)(c^2-a^2)$