Variance of filtered white noise
The output process is clearly zero-mean because the LTI system cannot add a mean to the zero-mean input process. The variance of the filtered process is given by
$$E\{Y^2\}=\frac{N_0}{2}\int_{-\infty}^{\infty}|h(t)|^2dt\tag{1}$$
which results in $3A^2N_0/2$.