Finding $P(W\le 1)$ given moment generating function of $W$

Question

Suppose that the moment generating function is given for a random variable $W$:`

$$M_w (t)=e^{-3t+3e^t }(0.75+0.25e^t )^4$$

The mgf seems to be a mixture of both a binomial and Poisson distribution. How would I find the probability of $P(W\le 1)$?

here is a image of the expression fully written out:

Answer 1

Hint: After correcting the typo, show that $W=X+Y$ where $X \sim \text{Poisson}(3)$ and $Y \sim \text{Binomial}(4, 0.25)$ are independent. Then, $$P(W \le 1) = P(W=0) + P(W=1) = P(X=0, Y=0) + P(X=1, Y=0) + P(X=0, Y=1).$$