= Solution
Write
$$
\frac1{|\mathbf x-\mathbf y|}
=\frac1{|\mathbf x|}
\left(1-2\frac{\mathbf x\mathbin{\cdot}\mathbf y}{|\mathbf x|^2}
+\frac{|\mathbf y|^2}{|\mathbf x|^2}\right)^{-1/2}.
$$
Using the <Taylor series>
$$
(1+s)^{-1/2}=1-\frac12s+\frac38s^2+O(s^3)
$$
and retaining terms through second order in $|\mathbf y|/|\mathbf x|$ gives
$$
\boxed{
\frac1{|\mathbf x-\mathbf y|}
=\frac1{|\mathbf x|}
\left[
1+\frac{\mathbf x\mathbin{\cdot}\mathbf y}{|\mathbf x|^2}
+\frac{3(\mathbf x\mathbin{\cdot}\mathbf y)^2-|\mathbf x|^2|\mathbf y|^2}
{2|\mathbf x|^4}
+O\!\left(\frac{|\mathbf y|^3}{|\mathbf x|^3}\right)
\right]}.
$$
This is the beginning of the <multipole expansion>.
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