jmc

Multiplying both sides by $2x+37$ gives $$\begin{array}{rl}6x^2+111x+1& =(2x+37)(3x+1)\\ & =2x(3x+1)+37(3x+1)\\ & =6x^2+2x+111x+37\\ & =6x^2+113x+37\end{array}$$So, we have $$6x^2+111x+1=6x^2+113x+37.$$Subtracting $6x^2$ from both sides gives $111x+1=113x+37$. Rearranging this equation gives $2x=-36$, from which we find $x=\boxed{-18}$.