variance of the product of two samples with awgn
There is a paper, which is more general than Gaussian distributions.


  Bohrnstedt, George W., and Arthur S. Goldberger. "On the exact
  covariance of products of random variables." Journal of the American
  Statistical Association 64.328 (1969): 1439-1442.


https://www.jstor.org/stable/pdf/2286081.pdf?casa_token=m1_rSG4hiwIAAAAA:sGjisD6dekanCat5EBGl0yvdn1EJ5LY783fq4VKdxNhd0RA4z4izkcyDJ6yU7MylDke_qD335SWUNu1gJ5opTzXFWIVQWDrjoMvxcFgvMBEgG6r8En0W

Their equation (9)
$$
 V(xy) = E^2(x)V(y) + E^2(y)V(x) + V(x)V(y). \quad (9)
$$
reduces to your result