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