Let $Z\sim N(0,1)$ and $Y=a+bZ+cZ^2$. I want to compute the variance of $Y$. This is what I did: $$\operatorname{Var}(Y)=0+b^2\operatorname{Var}(Z)+c^2\operatorname{Var}(Z^2)=b^2+c^2\operatorname{Var}(Z^2)$$ To get $\operatorname{Var}(Z^2)$, I tried to use the definition $\operatorname{Var}(Z^2)=\mathbb{E}[Z^4]-\mathbb{E}[Z^2]^2$ But im having with this part. If this was a odd for example $\mathbb{E}[Z^3]$ you can say that because of the symmetry of the normal distribution $\mathbb{E}[Z^3]=0$, but in this is pair.
Thank you