Example 8 Find all positive integers $a, b$ such that
$$4^{a}+4 a^{2}+4=b^{2}$$
If $(a, b)$ is a pair of positive integers satisfying (1), then $b^{2}$ is even, and $b^{2}>4^{a}$, so $b$ is even, and $b>2^{a}$, hence $b \geqslant 2^{a}+2$. Therefore,
$$4^{a}+4 a^{2}+4=b^{2} \geqslant\left(2^{a}+2\right)^{2}=4^{a}+4 \cdot 2^{a}+4$$

we have $a^{2} \geqslant 2^{a}$, which implies $a \leqslant 4$ (by induction on $a$, we can prove that when $a \geqslant 5$, $a^{2}<2^{a}$).
Substituting $a=1,2,3,4$ into equation (1), we find that all positive integer solutions of the equation are $(a, b)=$ $(2,6)$ or $(4,18)$.