Polynomial pencil with four square members

Let $u,v\in \mathbb{C}[t]$ be coprime polynomials. If $\alpha u+\beta v$ is a square for four distinct $(\alpha :\beta )$, then $u,v$ are constant. For independent $u,v$, the four square members $L_j=s_j^2$ are pairwise coprime polynomials. Let $d$ and $e$ be their maximal and minimal degrees; at least three have degree $d$. For independent members of degrees $e,d$, the nonzero polynomial $W=L_k^{\prime }{L}_l-L_k{L}_l^{\prime }$ has degree at most $d+e-1$. Each $s_j$ divides $W$, giving $\mathrm{deg}W\ge (3d+e)/2>d+e-1$, a contradiction. The dependent case follows directly from coprimality of polynomials.