10.8.6 Series with complex terms

Problem: Expand \( f(z) \) into Laurant series (Taylor) in a given ring or neighborhood of a given point (in the latter case, indicate the area of convergence of the resulting series). \[ f(z)=\frac{z-4}{z^{2}(z+4)}, \quad 1<|z-1|<5 . \]