Hey theconman12345.

You can use the moment generating function of the sum of two independent normal distributions is a normal and you can use the fact that E[X+Y] = E[X] + E[Y] and Var[X+Y] = Var[X] + Var[Y] to show you get the mean and variance added when they are independent.

The MGF proof is obtained by considering that if X and Y are independent, then E[e^(t{X+Y})] = E[e^tX]E[e^tY] = MGF_x(t)*MGF_y(t) and you can take it from there.