Supposing we know $f_1(x),f_2(x),f_3(x),f_4(x),f_5(x),f_6(x),f_7(x)\in\Bbb F_q[x]$ in $$f_7(x)=f_1(x)(\alpha f_2(x)+f_3(x))+f_4(x)(\beta f_5(x)+f_6(x))$$ where $\alpha(x),\beta(x)\in\Bbb F_q$ are unknown but of degree at most $k$ (a constant). Then can there be more than one pair of $\alpha(x),\beta(x)\in\Bbb F_q$ which satisfy $f_7(x)$ (note $\mathsf{deg}(f_3(x))\leq\mathsf{deg}(f_2(x))$ and $\mathsf{deg}(f_6(x))\leq\mathsf{deg}(f_5(x))$ may be assumed if needed)?
How to find at least one pair without trialing all possible values of $\Bbb F_q$ in preferrably $O((\log q)^c)$ steps for some $c>0$?
Assuming a solution exists is it as simple as choose an arbitrary polynomial $\alpha(x)$ and find $\beta(x)$? And so are there are always $q^{k+1}$ solutions even if one exists?