jmc

Since the graph passes through the points $(-2,0)$ and $(4,0),$ the equation is of the form $a(x+2)(x-4).$

The graph has a maximum, and this maximum value occurs at the average of $-2$ and 4, namely $x=\frac{-2+4}{2}=1.$ But $a+b+c$ is exactly the value of $y=a{x}^2+bx+c$ at $x=1,$ so $a+b+c=\boxed{54}.$