E[X] is just a number, so you have to work out the variance of a constant times N. That is standard: Var(kY)=k^2.Var(Y) for k constant and Y an r.v.
Don't forget that Var (Y)= E(Y^2) - E(Y)^2 as well, when you're doing things like this. So if U and V are independent
Var(UV)= E(U^2V^2) - E(UV)^2 = E(U^2)E(V^2) - E(U)^2E(V^2)
which can be related, albeit messily, to Var(U) and Var(V).