The MGFs are unique. So, once you noticed that
is a binomial with n=5 and p=.7.
Then the mean is just np=(5)(.7)=3.5 and the variance is npq....
There's no need to differentiate.
For the third MGF, you need to go back to the definition of our MGF.
I always mention these.
Now just match up terms. The coefficients are the probabilities and that's why they sum to one, and .
And the constants in front of the t in the exponent are just the realizations of x.