 Solution

If $\alpha =-b/(2a)$ is a repeated root, then $A=0$ and both coefficients of $\beta$ in $B$ vanish, so $B=0$ for every $\beta$. Choose $\beta \ne \alpha$, giving $C\ne 0$ and hence $v_{\eta \eta }=0$. For $(s+1{)}^2$, take $\xi =y-x$, $\eta =y$, and obtain