= Solution
For a <Runge-Kutta method>, the <stability function> is $R(z)=1+z b^T(I-zA)^{-1}e$. Solving the stage equations on the <Dahlquist test equation> gives
$$
R(z)=\frac{1+z/2+z^2/12}{1-z/2+z^2/12}
=\frac{z^2+6z+12}{z^2-6z+12}.
$$
The denominator has <roots of a polynomial> $3\pm i\sqrt3$, both in the open right half-plane. If $z=x+iy$, direct expansion gives
$$
|z^2-6z+12|^2-|z^2+6z+12|^2=-24x(|z|^2+12).
$$
Since $|z|^2+12>0$, the <modulus> of $R$ is at most one exactly when $x\leq0$. Thus
$$
\boxed{\mathcal S=\{z\in\mathbb C:\operatorname{Re}z\leq0\}.}
$$
\b[The method is <A-stable>.] Its <stability function> has unit <modulus> on the imaginary axis and tends to one as $z\to-\infty$, so it is not <L-stable>.
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