1. Let $p$ be a real number, and the graph of the quadratic function $y=x^{2}-2 p x-p$ intersects the $x$-axis at two distinct points $A\left(x_{1}, 0\right), B\left(x_{2}, 0\right)$.
(1) Prove that $2 p x_{1}+x_{2}^{2}+3 p>0$;
(2) If the distance between points $A$ and $B$ does not exceed $|2 p-3|$, find the maximum value of $p$.
(2000, National Junior High School Mathematics Competition)
(Tip: Extreme method. According to the relationship between roots and coefficients. Answer: $\frac{9}{16}$. )