This belongs to the class of "*nonsymmetric algebraic Ricatti equations*", which can be solved following the method explained in <A HREF="http://epubs.siam.org/doi/pdf/10.1137/S0895479800375680">Nonsymmetric algebraic Riccati equations and Wiener–Hopf factorization for M-matrices</A> (2001) and in <A HREF="https://mathoverflow.net/users/1898/federico-poloni">Federico Poloni's</A> paper on <A HREF="https://arxiv.org/abs/1004.1500">Quadratic vector equations</A> (2010).