Let $$Y =\begin{pmatrix} Y_1 \\ Y_2 \end{pmatrix} ∼ N(0, \Sigma) \quad \Sigma=\begin{pmatrix} \sigma_{11} & \sigma_{12}\\ \sigma_{21} & \sigma_{22} \end{pmatrix}$$$$Y =\begin{pmatrix} Y_1 \\ Y_2 \end{pmatrix} \sim N(0, \Sigma) \quad \Sigma=\begin{pmatrix} \sigma_{11} & \sigma_{12}\\ \sigma_{21} & \sigma_{22} \end{pmatrix}$$
Show that $$\Bbb E(Y_1^2Y_2^2)=\sigma_{11}\sigma_{22} + 2\sigma_{12}^2. $$$$\mathbb E(Y_1^2Y_2^2)=\sigma_{11}\sigma_{22} + 2\sigma_{12}^2. $$ I tried to turn this in all the ways but I need $Var(Y_1Y_2)$$\mathbb {Var}(Y_1Y_2)$ that I don't have. In the solution I was given, it only uses the fact that "the distribution is Gaussian".
