\[ \begin{split} &\lim_{z \to -2}\frac{z(z^2+4z+4)}{(z+2)(z-3)}=\\[16pt] &=\lim_{z \to -2}\frac{z(z+2)^2}{(z+2)(z-3)}=\\[16pt] &=\lim_{z \to -2}\frac{z(z+2)}{z-3} =\\[16pt] &=\frac{0}{-5}=0 \end{split} \]