Distribution of the combination of square and the product of Gaussian random variables

Question

What is the distribution of the following expression? $$ \sum_{i=1}^n a_i x_i^2 + \sum_{i \neq j; i,j=1}^n b_{ij} x_i x_j $$ where $x_i$ for $i$ in the range from 1 to $n$ are i.i.d. samples from $\mathcal{N}(0,1)$, $a_i$, and $b_{ij}$ are real constants.

Answer 1

Looks like a quadratic form $x^\top Mx$, where $x\sim \mathcal N(0,I_n)$ and $M$ is symmetric.

If $M$ is an idempotent matrix of rank $m$ then see this link $$x^\top Mx\sim\chi^2(\text{tr}(M))$$ However, if $M$ is not idempotent, it is not a well-known distribution. For further properties of this quadratic form, see Variance of quadratic form for multivariate normal distribution.