= Solution
Put $\mathbf J=\mathbf S_1+\mathbf S_2$. <Angular momentum addition> gives $j=0,1,2,3$, and the <spin-dot-product eigenvalue> is
$$
\frac{\hbar^2}{2}\left[j(j+1)-2\frac32\frac52\right].
$$
Thus the <spin-three-halves pair spectrum> is
$$
\boxed{\frac{\hbar^2}{4}(-15,-11,-3,9),\quad\text{with multiplicities }1,3,5,7.}
$$
For identical <fermions>, <exchange parity of two coupled identical spins> makes $j=0,2$ spin-antisymmetric and $j=1,3$ spin-symmetric. Corresponding spatial exchange symmetries make the full state antisymmetric; without a specified spatial state, none of these <total-spin sectors> is excluded.
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