8.1. The real numbers $p$ and $q$ simultaneously satisfy the relations:

$$
2 p^{2}-3 p-1=0, \quad q^{2}+3 q-2=0, \quad p \cdot q \neq 1
$$

Determine the numerical value of the expression

$$
E(p, q)=\frac{23 p^{4}+125 q^{4}}{17 p^{4}-q^{4}}
$$
Solution. According to the relationships in the statement, we observe that $p q \neq 0$. Then the following equalities are correct:

$$
\begin{aligned}
q\left(2 p^{2}-3 p-1\right)+p\left(q^{2}+3 q-2\right)=0 & \Leftrightarrow 2 p^{2} q-3 p q-q+p q^{2}+3 p q-2 p=0 \Leftrightarrow \\
p q(2 p+q)-(2 p+q)=0 & \Leftrightarrow \quad(p q-1) \cdot(2 p+q)=0
\end{aligned}
$$

Since $p \cdot q \neq 1$, it follows that the equalities $2 p+q=0 \quad \Leftrightarrow \quad q=-2 p$ are correct. Then we obtain

$$
E(p, q)=\frac{23 p^{4}+125 \cdot(-2)^{4} p^{4}}{17 p^{4}-(-2)^{4} p^{4}}=\frac{(23+125 \cdot 16) \cdot p^{4}}{(17-16) \cdot p^{4}}=2023
$$

According to the conditions in the statement, the numerical value of the expression $E(p, q)$ is equal to 2023.