Given $(X,Y)$ multivariate normal distribution with $\bar\mu=0$ show that $E[X^2Y^2] = 2(E[XY])^2+E[X]^2E[Y]^2$

Question

Given $(X,Y)$ multivariate normal distribution with $\bar\mu=0$ show that $E[X^2Y^2] = 2(E[XY])^2+E[X]^2E[Y]^2$.

The question came with a clue saying to aproach the question with conditional expectation.

I have tried solving the question using the clue but I got stuck pretty early

$E[X^2Y^2]=E[E[X^2Y^2|Y]]=E[Y^2E[X^2|Y]]$

not knowing how to compute $E[X^2|Y]$ because I dont even know if there is a PDF.

I solved the question using a different approach using the 4th-order central moment

$E[X^2Y^2]=E[XXYY]=E[XX]E[YY]+E[XY]E[XY]+E[XY]E[XY]=2(E[XY])^2+E[X^2]E[Y^2]$

but i would still like to know how to solve the question using conditional expectation.