Answer
$2Co(NO_{3})_{3}(aq)+3(NH_{4})_{2}S(aq)\ -->\ Co_{2}S_{3}(s)+6NH_{4}NO_{3}(aq)$
Work Step by Step
Add a 3 to the $(NH_{4})_{2}S$ to balance out the sulfur on the $Co_{2}S_{3}$. Then add a 6 to the $NH_{4}NO_{3}$ to balance out the ammonium on the $(NH_{4})_{2}S$. Finally add a 2 to the $Co(NO_{3})_{3}$ to balance the 2 cobalt and 6 $(NO_{3})^{-}$.
