homework and exercises - Calculating imaginary part of a loop diagram using cutting rules for phi^4 theory

I'm trying to calculate the imaginary part of this diagram

in $\phi^4$ theory, using the optical theorem, and I'm having trouble.

The cutting rules seem to suggest that this diagram is equal to $$ (-i \lambda)^2 \int \frac{d^4 l}{(2\pi)^4} \frac{i}{(k+q+l)^2 - m^2} \frac{i}{(l-p)^2 - m^2}$$ $$= -(-i \lambda)^2 \int \frac{d^4 l}{(2\pi)^4} (-2\pi i)\delta \left ((k+q+l)^2 - m^2 \right ) (-2 \pi i)\delta \left ((l-p)^2 - m^2 \right )$$ wherein $k+q$ and $p$ are the ingoing momenta, $p+q$ and $k$ are the outgoing momenta, and $l$ is the loop momentum - i.e., just the product of two vertices, times an integral over a couple of delta functions.

Edit: Assuming that what precedes is correct, the difficulty comes in the evaluation of those delta functions. The next step seems to be to expand them in terms of the root expansion for delta functions, $$ \delta(f(x)) = \sum_{y | f(y) = 0} \frac{\delta(x-y)}{\left |f'(y) \right|} $$ which just illustrates the problem: in general, the roots of the arguments of the delta functions will be complex. Taken naïvely, the integral over the deltas will yield 0 because the roots of their arguments won't lie on the real line. This article discusses the difficulty, but isn't sufficiently explicit for me to implement the calculational technique. Once again assuming that the equation I got from the cutting rules is correct, how do I evaluate the integrals of those deltas?

Alternatively, what I'd really like is a completely-worked, step-by-step example calculation. If anybody can provide one or point me to one, that would be great.

This related question may also be of interest.