I have a system of $n \times 1$ equations $$ 0 = A\,vec(xx^t) + B x + C $$ where

• $x$ is a $n \times 1$ vector of unknowns

• $x^t$ means transpose

• $vec$ means $xx^t$ has been vectorized so has dimension $n^2 \times 1$

• $A$ is a known matrix with dimensions $n \times n^2$

• $B$ is a known matrix with dimensions $n \times n$

• $C$ is a known vector of dimension $n \times 1$

I can solve these problems using a nonlinear solver. However, I am trying to find out if there are any theoretical results on how to solve this class of problems. I can find a lot of work on solving matrix quadratic equations. However, I can't find anything that has specifically this form, and I cannot figure out if I can rewrite this system in a way that is equivalent to other matrix quadratic problems I come across.