Note that the mgf of a random variable \( X \) is defined by

\( M_X(t) = E[e^{tX}] \)

and mathematically speaking it is a Laplace transform of the corresponding pmf/pdf. Now you are given the mgf, and want to find the corresponding pmf/pdf, so generally speaking you would like to do an inverse-Laplace transform. There are table for this for many common functional form.

However, your mgf is easy enough to recognize. Think about how do you calculate the mgf for a discrete distribution, say a Binomial distribution. Then you can match the corresponding probabilities and support points. Also note that the Laplace transform is unique so you will have a unique solution.