Example 5. Determine the value of the natural number $n$, such that the two roots of the quadratic equation $2 x^{2}-8 n x+10 x-n^{2}+35 n-76$ $=0$ are prime numbers, and find these two roots. (1990, Shanxi Province Junior High School Mathematics Competition)
Solve: From the given equation, we get $x_{1}+x_{2}=4 n-5$. Since $x_{1}$ and $x_{2}$ are both prime numbers, and $4 n-5$ is an odd number, it follows that one of $x_{1}$, $x_{2}$ must be even, and the only even prime number is 2. Without loss of generality, let $x_{1}=2$, substituting into the original equation yields $n_{1}=16, n_{2}=3$, the corresponding roots are $x_{1}=2, x_{2}=57$ or $x_{1}=2, x_{2}=5$.