= Solution
For each nonzero <period lattice> point $\omega$, the <binomial series> gives
$$
\frac1{(z-\omega)^2}-\frac1{\omega^2}
=\sum_{k=1}^{\infty}(k+1)\frac{z^k}{\omega^{k+2}}.
$$
Pairing $\omega$ with $-\omega$ cancels every odd power. Hence the <Laurent series> of the <Weierstrass elliptic function> is
$$
\wp(z)=\frac1{z^2}
+3z^2\sum_{\omega\ne0}\omega^{-4}
+5z^4\sum_{\omega\ne0}\omega^{-6}+O(z^6)
=\frac1{z^2}+\frac{g_2}{20}z^2+\frac{g_3}{28}z^4+O(z^6),
$$
which is the stated <Laurent coefficients of the Weierstrass elliptic function>[coefficient formula].
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