jmc

Multiplying both sides by $x+3,$ we have $-x^2(x+3)=3x+1,$ or $-x^3-3x^2=3x+1.$ Thus, $$x^3+3x^2+3x+1=0.$$We recognize the left-hand side as the expansion of $(x+1{)}^3,$ so $$(x+1{)}^3=0.$$This forces $x+1=0,$ so $x=\boxed{-1},$ which is the only solution.