How to take the variance of a second order expansion? $\text{Var}\left[aX+bY+cXY+mX^2+nY^2\right]$
Let say we have 5 real-valued constant parameters $\{a,\ b,\ c,\ m,\ n\}$, and two random variables $X$ and $Y$ such their means values are $\mu_x$ and $\mu_y$, their variances are $\sigma_x^2$ and $\sigma_y^2$, and they are correlated with coefficient $\rho_{xy}$.
I want to know how to calculate the following variance and which value it has based on the previous parameters: $$\text{Var}\left[aX+bY+cXY+mX^2+nY^2\right]$$
- Could this be done if "nothing" is known about the probability distribution of each random variable (maybe different distributions for each one)? (but considering that it is true that each one has finite parameters $\mu_i$, $\sigma_i^2$, and $\rho_{ij}$).
- If point (1) is not possible: Could this be answer if $X$ and $Y$ are normally distributed? (I found the Isserlis' theorem and Stein's lemma, but I got stuck anyway).
I aiming to find a formula I could directly use, so please explain if your results are based in any assumptions (unfortunately, independence is not applicable for the result I am looking for).