1. Let the polynomial $p_{k}(x)=\left(\cdots\left(\left((x-2)^{2}-2\right)^{2}-2\right)^{2}-\cdots-2\right)^{2}$, where $k$ is any given positive integer, find the coefficient of $x^{2}$ in $p_{k}(x)$.
1. From the problem, we know that the polynomials $P_{k}(x), P_{k-1}(x)$ satisfy $P_{k}(x)=\left(P_{k-1}(x)-2\right)^{2}, P_{k}(0)=4$. Let $P_{k}(x)=\cdots+A_{k} x^{2}+B_{k} x+4$. Then, after calculating and rearranging $P_{k}(x)-2$ squared, we get
$$
P_{k}(x)=\cdots+\left(B_{k}+4 A_{k}\right) x^{2}+4 B_{k-1} x+4 \text {. }
$$

Thus, $B_{k}=4 B_{k-1}, A_{k}=B_{k-1}^{2}+4 A_{k-1}, B_{1}=-4, A_{1}=1$.
From the above recursive relations, we derive $B_{k}=-4^{k}, A_{k}=\frac{1}{3}\left(4^{2 k-1}-4^{k-1}\right)$.