Precalculus · Superchallenge problem

The rational function $f(x)=\dfrac{x^3+ax^2+bx+c}{x^2-4}$ has a hole (removable discontinuity) at $x=2$ whose $y$-coordinate is $\dfrac52$, a vertical asymptote at $x=-2$, and the slant asymptote $y=x+3$. Find $a+b+c$.