I don't think there is a general theorem for that one (well, it is a kind of convolution, but not the usual one).

The method is the following: using the independence, we can write

and you know (substitute with appropriate mean and variance), so you are reduced to computing (with appropriate a and b). Write it as an integral and you probably know how to compute that (complete the square,...). The computation is easy if the mean is 0 (use the fact that the density of the Gaussian integrates to 1).