jmc

To get polynomials on both sides, we multiply both sides of the equation by $(x+2{)}^4$. This gives us, $$5x^4-8x^3+2x^2+4x+7=a(x+2{)}^4+b(x+2{)}^3+c(x+2{)}^2+d(x+2)+e.$$Since the two new polynomials are equal for all $x>0$ (an infinite number of points), they must be equal for all $x$.

If we plug in $x=-1$, the right hand side becomes $a+b+c+d+e$, which is what we're looking for! Plugging $x=-1$ into both sides gives us: $$a+b+c+d+e=5(-1{)}^4-8(-1{)}^3+2(-1{)}^2+4(-1)+7=\boxed{18}.$$