Use the hint.
SO, plug in into the MGF of your gamma distribution.
Suppose that Θ is a random variable that follows a gamma distribution with parameters λ and α, where α is an integer, and suppose that, conditional on Θ, Χ follows a poisson distribution with parameter Θ. Find the unconditional distribution of α + Χ. (Hint: Find the mgf by using iterated conditional expectations)