Solve quadratic vector equation, with variable hidden inside scalar

Question

Let $\vec{f}$ $m\times 1$ unknown vector, given $n\times 1$ vector $\vec{F}$, $n\times m$ matrix A ($n<m$), nonzero vector $\vec{v}$ from the nullspace of A ($Av=0$), non-invertible symmetric square matrix $B$ and $m\times 1$ basis vectors $\vec{e}_i,\vec{e}_j$ ($1's$ only at the indeces $i$ and $j$). Solve the following indeterminate equation in terms of $\vec{f}$: $$AB(\vec{f}-\vec{e}_i^T\vec{f}\vec{e}_j^T\vec{f}\vec{v})=\vec{F}.$$ More generally, can such equations be solved when the scalar includes the variable in higher degrees? I.e. $$AB(\vec{f} - \underbrace{\vec{e}_i^T\vec{f}\vec{e}_j^T\vec{f}...\vec{e}_k^T\vec{f}}_{k \: repetitions \:of \: f}\vec{v})=\vec{F}.$$