Example 10 Given that $b$ and $c$ are integers, the equation $5 x^{2} + b x + c = 0$ has two roots that are greater than -1 and less than 0. Find the values of $b$ and $c$.
Solution: According to the graph of the quadratic function $y=5 x^{2}+b x+c$ and the given conditions:
When $x=0$, $5 x^{2}+b x+c>0$, we have $c>0$;
When $x=-1$, $5 x^{2}+b x+c>0$, we have $b^{2} \geqslant 20 c, c<5$.
If $c=1$, then from (2) and (4) we get $0<b<$ ú H. $b^{2} \equiv$ 20 , we get $b=5$;

If $c=2$, then $0<b<$ ? and $b^{2} \geqslant 40$, no integer solution;

If $c=3$, then $0<b<8$ and $b^{2} \geqslant 60$, no integer solution;

If $c=4$, then $0<b<9$ and $b^{2} \geqslant 80$, no integer solution.
Therefore, the values of $b$ and $c$ are $b=5, c=1$.