Example 1. If $p, q$ are both natural numbers, and the two roots of the equation $p x^{2}-$ $q x+1985=0$ are both prime numbers, then what is the value of $12 p^{2}+q$? (85 Beijing Mathematics Competition Question)
Since $1985=5 \times 397$, and the two roots $x_{1}, x_{2}$ are both prime numbers, and $x_{1} x_{2}=\frac{1985}{p}=\frac{5 \times 397}{p}$, thus $p=1$, then $x_{1}, x_{2}$ are 5 and $397, x_{1}+x_{2}=\frac{q}{p}$ $=q=402$, therefore $12 p^{2}+q=12+402=414$.