Example 7 Let $p_{k}(x)=(\underbrace{\left.\left(\cdots\left((x-2)^{2}-2\right)^{2}-2\right)^{2}-\cdots-2\right)^{2}}_{k \text { brackets }}, k$ is any given natural number, try to find the coefficient of the $x^{2}$ term in $p_{k}(x)$.
Analysis Direct expansion will obviously lead to a difficult situation, which is not advisable. Observing the characteristics of the expression of $p_{k}(x)$ inspires us to establish a recursive relationship to solve the problem.
Solution From the problem, we have $p_{k}(x)=\left(p_{k-1}(x)-2\right)^{2}, p_{k}(0)=4$. Let $p_{k}(x)=\cdots+A_{k} x^{2}+B_{k} x+4$, then
$$
\begin{array}{c}
p_{k-1}(x)=\cdots+A_{k-1} x^{2}+B_{k-1} x+4, \\
p_{k}(x)=\cdots+\left(B_{k-1}^{2}+4 A_{k-1}\right) x^{2}+4 B_{k-1} x+4 .
\end{array}
$$

Therefore,
$$
B_{k}=4 B_{k-1}, A_{k}=B_{k-1}^{2}+4 A_{k-1}, B_{1}=-4, A_{1}=1 \text {. }
$$

From the above recursive relationship, we solve to get
$$
B_{k}=-4^{k}, A_{k}=\frac{4^{2 k-1}-4^{k-1}}{3} .
$$

For some special numerical structures, using the method of generalizing special values, transitioning to polynomial research, can achieve the effect of simplifying complexity.