= Solution
The canonical <Kähler potential> gives $e^K=e^{|z|^2}$ and inverse <Kähler metric> one. Substituting the <superpotential> and its <Kähler covariant derivative of a superpotential> into the <supergravity F-term potential> yields
$$
\boxed{V(z)=|m|^4e^{|z|^2}\left(\left|1+\bar z(z+\beta)\right|^2-3|z+\beta|^2\right).}
$$
For real $m$, the prefactor is simply $m^4$; the modulus form also covers a complex phase. To keep both scalar directions explicit, write $z=x+iy$ and $P=1+x^2+y^2+\beta x$. Then
$$
V=|m|^4e^{x^2+y^2}H(x,y),\qquad
H=P^2+\beta^2y^2-3\{(x+\beta)^2+y^2\}.
$$
The negative term is essential: unlike the global <F-term scalar potential>, the <supergravity F-term potential> need not be nonnegative. This is why cancelling the <cosmological constant> does not force the <auxiliary field> to vanish.
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