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 $(2,0)$ and $(-1,0)$. If $a$ is a positive integer, which of the following could be the value of $a+b+c$?