The function $g$ is defined by $g(x)=p{x}^2+qx+r$, where $p$, $q$, and $r$ are constants. The graph of $y=g(x)$ in the $xy$-plane passes through the points $(5,0)$ and $(-1,0)$. If $p$ is an integer greater than $2$, which of the following could be the value of $p+q$?