There is a certain method to go about doing this. I remember the start and finish, but not the intermediate steps.

Let: X ~ Poisson( $\displaystyle \lambda$ )

Find: $\displaystyle E[X(X-1)]$

Or something like it. It had to do with the fact that E[X] in this case = $\displaystyle \lambda$

$\displaystyle E[X^2 - X]$

$\displaystyle = E[X^2] - E[X]$

... ... ... ...

$\displaystyle = \lambda + \lambda ^2 - \lambda$

$\displaystyle = \lambda ^2$