The function $f$ is defined by $f\left(x\right)=a{x}^2+bx+c$, where $a$, $b$, and $c$ are constants. The graph of $y=f\left(x\right)$ in the xy-plane passes through the points $\left(7,0\right)$ and $\left(-3,0\right)$. If $a$ is an integer greater than $1$, which of the following could be the value of $a+b$?