= Solution
The <square lattice> has <coordination number of a lattice> $z=4$. The mean-field decoupling is
$$
S_iS_j\simeq S(S_i+S_j)-S^2.
$$
Accounting for the factor $1/2$ in the directed-neighbour sum gives the <Hamiltonian> in the <mean-field approximation>:
$$
H_{\mathrm{MF}}=2JNS^2-(4JS+B)\sum_iS_i.
$$
The constant term is necessary to correct the double counting of the interaction. Let $h=4JS+B$. The one-site <partition function> is
$$
Z_1=\sum_{s=-1,0,1}e^{\beta hs}=1+2\cosh(\beta h),
$$
so $Z_{\rm MF}=e^{-2\beta JNS^2}Z_1^N$. The <Helmholtz free energy> is therefore
$$
\boxed{F_{\mathrm{MF}}(S,T,B)=N\left[2JS^2-\frac1\beta\log\{1+2\cosh[\beta(4JS+B)]\}\right].}
$$
The equilibrium <mean-field approximation> is obtained by minimizing this expression over the allowed $S$. The $s=0$ state supplies the extra $1$ in $Z_1$; omitting it would instead describe the two-state <Ising model>. This is the <Blume-Capel mean-field transition at zero single-ion anisotropy>.
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