= Solution
In the original coordinate, the possible zero-energy <vacuum expectation values> for $g\ne0$ are
$$
\boxed{v_\pm=\frac{-m\pm\sqrt{m^2-2g\kappa}}{g}.}
$$
They coincide when the <discriminant> vanishes. With real parameters and a nonnegative <discriminant> these are real; otherwise the allowed <complex scalar> values must be retained.
In the coordinate $u=\varphi-s$ used to remove the linear term, the same vacua are
$$
\boxed{\langle u\rangle=0\quad\text{or}\quad-\frac{2M}{g}.}
$$
The nonzero branch exists in this shifted coordinate when $M\ne0$. Let $w=-2M/g$ be that branch and write $u=w+\widetilde\varphi$. Equivalently, in the original coordinate write $\varphi=v+\widetilde\varphi$ with $v=s+w$. The physical fluctuation field is the same. For $g=0,m\ne0$ there is the unique vacuum $v=-\kappa/m$; for the constant and pure-linear exceptional branches the scalar is a flat direction, with the <supersymmetry> status already distinguished in part (c).
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