I am struggling on how to compute the variance of a variance estimator $\hat{\sigma}^2$, the problem is stated below. A hint is provided which says that the fourth moment of the expectation of a random variable equals $E[X^4] = 3\sigma^4$:
Show that $\hat\sigma^2 = \frac{1}{N+2}\sum_{n=0}^{N-1}(x[n]-\mu_x)^2$ has:
$$ \text{var}(\hat\sigma^2) = \frac{2(\sigma^2)^2N}{(N+2)^2} $$
Any lead on the right direction would be very much appreciated!