= Solution
For $K=|\phi|^2$,
$$
D_\phi W=m^2[1+\phi^*(\phi+\beta)],
$$
and the <supergravity F-term potential> is
$$
V(\phi)=m^4e^{|\phi|^2}
\left(
\left|1+\phi^*(\phi+\beta)\right|^2
-3|\phi+\beta|^2
\right).
$$
The potential is invariant under complex conjugation. Writing $\phi=x+iy$, its derivative with respect to $y$ vanishes at $y=0$, so the extremisation equations consistently admit a real solution. On that slice,
$$
V(x)=m^4e^{x^2}
\left[(1+x^2+\beta x)^2-3(x+\beta)^2\right].
$$
Solved by gpt-5.6-sol high.
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