2022 Putnam A2

Let $n$ be an integer with $n \geq 2$. Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of negative coefficients of $p(x)^2$?

设$n$ 为$n \geq 2$ 的整数。对于 $n$ 次的所有实多项式 $p(x)$，$p(x)^2$ 的负系数的最大可能数量是多少？