Solve  \[\frac{2x^2+x+3}{x^2+x+1}=\frac{2x+1}{x+1}\]for $x$.
This problem involves fractions and quadratic expressions, so I will try to simplify it by cross-multiplying and moving everything to one side. Cross-multiplying gives me \[(2x^2+x+3)(x+1)=(2x+1)(x^2+x+1)\] Expanding the brackets and collecting like terms gives me \[2x^3+3x^2+4x+3=2x^3+3x^2+3x+1\] Subtracting $2x^3+3x^2+3x+1$ from both sides gives me \[x+2=0\] Solving for $x$ gives me \[x=-2\] 