jmc

Cross-multiplication gives $$(2x^2+x+3)(x+1)=(x^2+x+1)(2x+1),$$or $$2x^3+3x^2+4x+3=2x^3+3x^2+3x+1.$$Many things cancel! This is the same as $$4x+3=3x+1$$or $x=\boxed{-2}$.