= Solution
Only $\alpha>0$ needs consideration, by the preceding <zero-stability> analysis. On the <Dahlquist test equation>, all roots of the <amplification polynomial>
$$
P_z(\zeta)=\rho(\zeta)-z\sigma(\zeta)
$$
must satisfy the unit-disk root condition if the method is <A-stable>. Evaluate it at $\zeta=-1$:
$$
P_z(-1)=8+2\alpha z.
$$
Take a real negative $z<-4/\alpha$. Then $P_z(-1)<0$, whereas the leading coefficient $2+\alpha-z$ is positive and $P_z(\zeta)\to+\infty$ as $\zeta\to-\infty$. The <intermediate value theorem> gives a real root strictly below $-1$. Its numerical mode grows in modulus, contradicting <absolute stability> at this negative test parameter. Consequently
$$
\boxed{\text{no parameter gives both convergence and A-stability}.}
$$
This proof detects an unstable root directly; the second-order value alone would not exclude <A-stability>. Equivalently the limiting derivative polynomial has an exterior root $-\alpha-\sqrt{1+\alpha^2}$, explaining the instability for large negative $z$.
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