Note that its coefficients are symmetric and $x$ cannot equal $0$. Therefore dividing the give relation by $x^2$ leads to $$x^2 - 97 x + 2012 -\frac{97}{x}+\frac{1}{x^2}=0$$ Letting $y=x+\frac{1}{x}$ and noting $x^2 +\frac{1}{x^2} = y^2 - 2$ yield $$y^2-97y+2010=0\implies y=30, 67$$ Accordingly, $x=\boxed{15\pm 4\sqrt{14}, \frac{2}{67+\sqrt{4485}}, \frac{67+\sqrt{4485}}{2}}$.