The graph of a polynomial function $f$ has x-intercepts at $(-10,0)$, $(-4,0)$, and $(2,0)$. Function $g$ is defined by $g(x)=f(2x)$. The graph of $y=g(x)$ in the $xy$-plane has x-intercepts at $(a,0),(b,0)$, and $(c,0)$, where $a,b$, and $c$ are distinct constants. What is the value of $a+b+c$?