<s>[INST] sum of integer parts of roots 
If $\alpha,\beta,\gamma$ are real roots of the equation $\displaystyle \frac{1}{x}+\frac{1}{x-1}+\frac{1}{x-2} = \frac{11}{54}x^2.$ Then $\lfloor \alpha \rfloor + \lfloor \beta \rfloor +\lfloor \gamma \rfloor $  

$\bf{Attempt}$ From equation $$\frac{x^2-3x+2+x^2-2x+x^2-x}{x(x-1)(x-2)} = \frac{11}{54}x^2$$
So $$\frac{3x^2-6x+2}{x^3-3x^2+2x} = \frac{11}{54}x^2$$
$$162x^2-324x+108 = 11x^5-33x^4+22x^3$$
$$11x^5-33x^4+22x^3-162x^2+324x-108 =0$$
Could some help me how to solve it, thanks
 [/INST] HINT: use that $$\frac{1}{x}+\frac{1}{x-1}-\frac{1}{x-2}-\frac{11}{54}x^2=-\frac{(x-3) \left(11 x^4+22 x^2-96 x+36\right)}{54 (x-2) (x-1) x}$$
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