Using substitution $p = x + \frac{1}{x}$ to reduce $2x^4+x^3-6x^2+x+2=0$

Question

How do I use the substitution $p = x + \frac{1}{x}$ to show that the equation $$2x^4+x^3-6x^2+x+2=0$$ reduces to $2p^2+p-10=0$.

Now, I can do the problem in reverse, but when I try solving $p = x + \frac{1}{x}$ for $x$ and substituting, the resulting equation is too complicated and nothing like the form required.

Answer 1

HINT :

Dividing the both sides by $x^2$ gives $$2x^2+x-6+\frac 1x+\frac{2}{x^2}=0,$$ i.e. $$2\left(x^2+\frac{1}{x^2}\right)+\left(x+\frac 1x\right)-6=0.$$

Answer 2

Hint: divide the equation by $x^2$, then group terms and use the identity $(x+\frac{1}{x})^2 = x^2 + \frac{1}{x^2} + 2$.