jmc

Combining the fractions on the left gives $\frac{2t-2}{t+2}=3$. Multiplying both sides by $t+2$ gives $2t-2=3(t+2)$. Expanding the right side gives $2t-2=3t+6$. Subtracting $2t$ and 6 from both sides gives $t=\boxed{-8}$.