Assuming a r.v. $X$ with pdf $N(0,σ_x^2)$ and an independent r.v $Y$ with pdf $N(0,σ_y^2)$

for a set of positive constants $\alpha$, $\beta$, $\gamma$

what would be the distribution of the r.v.

$$Z= \alpha X^2 + \beta XY + \gamma Y^2$$

The problem is more complicated with nonzero means, so I would like to tackle it step by step