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 $(-5,0)$ and $(1,0)$. If $a$ is an integer less than $-2$, which of the following could be the value of $a-b$?