Example 6 Find the integer points on the curve $4 x=y^{2}+5 y+2(0 \leqslant y \leqslant$ 50 ).
Solution: The problem is to find the coordinates of integer points on the curve.
Obviously, $x=\frac{y^{2}+5 y+2}{4}(0 \leqslant y \leqslant 50)$.
Divide the integer $y$ into four categories: $4 n, 4 n+1, 4 n+2$ $(n=0,1,2, \cdots, 12), 4 n+3(n=0,1,2, \cdots$, 11), and use the elimination method for discussion.
When $y=4 n$, $x=4 n^{2}+5 n+\frac{1}{2}$;
When $y=4 n+1$, $x=4 n^{2}+7 n+2$;
When $y=4 n+2$, $x=4 n^{2}+9 n+4$;
When $y=4 n+3$, $x=4 n^{2}+11 n+\frac{13}{2}$.
When $y$ is $4 n$ and $4 n+3$, $x$ is not an integer, so the required integer points are $\left(4 n^{2}+7 n+2,4 n+1\right)$ and $\left(4 n^{2}+9 n+4,4 n+2\right)(n=0,1,2, \cdots, 12)$.