= Solution
The <self-consistency equation> requires that the mean spin computed using the single-site <Boltzmann distribution> equals the spin $S$ used to form the effective field. With $h=4JS+B$,
$$
\langle s\rangle_h=\frac{\sum_{s=-1,0,1}s e^{\beta hs}}{\sum_{s=-1,0,1}e^{\beta hs}}=\frac{e^{\beta h}-e^{-\beta h}}{1+e^{\beta h}+e^{-\beta h}}.
$$
Thus
$$
\boxed{S=\frac{2\sinh[\beta(4JS+B)]}{1+2\cosh[\beta(4JS+B)]}.}
$$
Equivalently, differentiating part (b) gives $N^{-1}\partial F_{\rm MF}/\partial S=4J[S-\langle s\rangle_h]$, so stationarity of the <Helmholtz free energy> in the <mean-field approximation> is exactly this condition. Among multiple <stationary points> one must select the stable <minimum>.
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