Can someone give me some direction on this? I tried something but I have no definitions or theorems to support whether what I did is correct. Thanks!

Let X~Normal( , ), Y~Gamma( , ), and X and Y are independent. The moment generating function of X and Y are

and , respectively.

Find the moment generating function of X+Y.

Find the moment generating function of 5X.

Find the moment generating function of 2 + Y.

Find the moment generating function of 2 + 3X + 4Y.

Thanks!