= Solution
Successive reflections in the two grounded planes require four charges:
$$
\begin{array}{c|c}
\text{position}&\text{charge}\\ \hline
(a,b,0)&q\\
(-a,b,0)&-q\\
(a,-b,0)&-q\\
(-a,-b,0)&q.
\end{array}
$$
Their total charge and <electric dipole moment> vanish. Applying the second-order expansion from part (a), the terms proportional to $a^2x^2$, $b^2y^2$, and $a^2+b^2$ also cancel, while the mixed terms add. Thus
$$
\boxed{
\Phi(\mathbf r)\sim
\frac{q}{4\pi\epsilon_0}\frac{12abxy}{r^5}
=\frac{3qabxy}{\pi\epsilon_0r^5}
},
\qquad r\gg a,b.
$$
This is an <electric quadrupole> potential.
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