 Negative-real instability of the symmetric two-derivative formula

On the Dahlquist test equation, the symmetric two-step two-derivative formula has amplification polynomial $F_z(w)=(z^2-7z+15)w^2-16zw-(z^2+7z+15)$. For every real $z<0$, its leading coefficient is positive and $F_z(-1)=2z<0$, whereas $F_z(w)\to +\infty$ as $w\to -\infty$. The intermediate value theorem therefore gives a real root below minus one. Thus the method is not A-stable, without any appeal to a multistep order barrier.