How can this expression be simplified? How do I factorize $$a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)?$$ 
I've tried it in different ways but failed. Wish some one could help solving it out.

Another proof. By Laplace expansion (along the last row):
$$ -\sum_{cyc}a^2(b-c) = \det\begin{pmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{pmatrix},\tag{1}$$
But the RHS is a Vandermonde matrix, whose determinant is well-known.
Gaussian elimination gives:
$$ D=\det\begin{pmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{pmatrix}=\det\begin{pmatrix} 1 & 0 & 0 \\ a & b-a & c-a \\ a^2 & b^2-a^2 & c^2-a^2 \end{pmatrix} \tag{2}$$
and by expanding along the first row and factoring out $(b-a)$ and $(c-a)$:
$$ D = (b-a)(c-a)\cdot \det\begin{pmatrix} 1 & 1 \\ b+a & c+a\end{pmatrix}=(b-a)(c-a)(c-b).\tag{3}$$
