<s>[INST] Quadratic substitution question: applying substitution $p=x+\frac1x$ to $2x^4+x^3-6x^2+x+2=0$ By using the substitution $p=x+\frac{1}{x}$, show that the equation $$2x^4+x^3-6x^2+x+2=0$$
reduces to $2p^2+p-10=0$.
I can't think of anything that produces a useful result, I tried writing p as $p=\frac{x^2+1}{x}$ and finding areas to substitute but have come with no progress. Could someone offer a slight hint on how to proceed?
Thanks
 [/INST] $$p=x+x^{-1}$$
$$p^2=x^2+2+x^{-2}$$
$$2x^4+x^3-6x^2+x+2=0$$
$$2x^2+x-6+x^{-1}+2x^{-2}=0$$
$$2(x^2+2+x^{-2})+(x+x^{-1})-10=0$$
$$2p^2+p-10=0$$
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