Determinant with Very Hard Algebra

The problem is

$\displaystyle

\det \left [ \begin{matrix} a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{matrix} \right ]

$

Sorry I'm new. That's the best I could write the problem. It's easy to find the determinant with expansion by minors but to get the simplified solution involves some sort of clever trick that I don't know of. Here's what I can get to.

$\displaystyle

abc(bc^2 - b^2c - ac^2 + a^2c + ab^2 - a^2b)

$

There are many ways to factor out terms from this.

$\displaystyle

abc[bc(c - b) + ac(a - c) + ab(b - a)] =

$

$\displaystyle

abc[b(c + a)(c - a) + c(a + b)(a - b) + a(b + c)(b - c)] =

$

$\displaystyle

abc[c^2(b - a) + b^2(a - c) + a^2(c - b)]

$

There are even more ways of course. None of these ways lead me to the simple solution:

$\displaystyle

abc(b - a)(c - a)(c - b)

$

Can someone explain how to factor this? Thanks.