22. Given the equations
$$
z^{2}=4+4 \sqrt{15} \mathrm{i} \text { and } z^{2}=2+2 \sqrt{3} \mathrm{i}
$$

the roots of these equations are the coordinates of the four vertices of a parallelogram in the complex plane. If the area \( S \) of the parallelogram can be expressed as \( p \sqrt{q} - r \sqrt{s} \left(p, q, r, s \in \mathbf{Z}_{+}, r, s \text{ are non-perfect squares} \right) \), then the value of \( p+q+r+s \) is ( ).
(A) 20
(B) 21
(C) 22
(D) 23
(E) 24
22. A.

The roots of the two equations are respectively
$$
\begin{array}{l}
z_{1}=\sqrt{10}+\sqrt{6} \mathrm{i}, z_{2}=-\sqrt{10}-\sqrt{6} \mathrm{i} ; \\
z_{3}=\sqrt{3}+\mathrm{i}, z_{4}=-\sqrt{3}-\mathrm{i} .
\end{array}
$$

Thus, the four vertices of the parallelogram are
$$
\begin{array}{l}
(\sqrt{10}, \sqrt{6}),(-\sqrt{10},-\sqrt{6}), \\
(\sqrt{3}, 1),(-\sqrt{10},-1) .
\end{array}
$$

Therefore, $S=6 \sqrt{2}-2 \sqrt{10}$
$$
\Rightarrow p+q+r+s=6+2+2+10=20 \text {. }
$$