It is clear that $x\ne 0$. Dividing both sides by $x^2$ yields $$x^2-97x+2012-\frac{97}{x}+\frac{1}{x^2}=0$$

Let $u=x+\frac{1}{x}$, then $x^2 +\frac{1}{x^2}=u^2-2$: $$(u^2-2)-97u+2012=0\implies u = 30, 67$$

Setting them back to $u=x+\frac{1}{x}$ gives the final results as $$x=15\pm 4\sqrt{14}, \frac{2}{67+\sqrt{4485}}, \frac{67+\sqrt{4485}}{2}$$