jmc

Cross-multiplying (which is the same as multiplying both sides by $r-2$ and by $r+1$) gives $$(r+3)(r+1)=(r-1)(r-2).$$Expanding the products on both sides gives $$r^2+3r+r+3=r^2-r-2r+2.$$which simplifies to $r^2+4r+3=r^2-3r+2$. Subtracting $r^2$ from both sides and collecting terms gives $7r=-1$, so $r=\boxed{-\frac{1}{7}}$.