$$\oint_\Gamma \frac{3z -1}{(z+2i)^3 z^2} dz$$
Here $\Gamma$ is the circle $|z| = 1/2$ traveling once counterclockwise
$$\oint_\Gamma \frac{\frac{3z -1}{(z+2i)^3}} {z^2} dz$$
$$= 2\pi i \frac{d}{dz} \frac{3z - 1}{(z+2i)^3} \big|_{z=0}$$
$$= 2\pi i \left(\frac{-24i - 12}{-64}\right)$$