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: $$a+b+c+d+e = 5(-1)^4 -8(-1)^3 + 2(-1)^2 + 4(-1) + 7 = \boxed{18}.$$