The function $f$ is defined by $f(x)=a{x}^2+bx+c$, where $a$, $b$, and $c$ are constants. The graph of $y=f(x)$ in the $xy$-plane passes through the points $(7,0)$ and $(-3,0)$. If $a$ is an integer greater than $0$, which of the following could be the value of $b+c$?