Simplification of special quartic polynomial

Question

Considering the quartic

$$p(x) = x^4+bx^3+cx^2+bx+1$$

Is it possible to convert this equation to a quadratic (bi-quadratic) polynomial?

Answer 1

Not sure if this is what youre looking for but here it goes

$$\begin{align} x^4+bx^3+cx^2+bx+1&=0\\ x^2+bx+c+{b\over x}+{1\over x^2}&=0\\ \left({x+\frac1x}\right)^2+b\left({x+\frac1x}\right)+c-2&=0\\ \left({x+\frac1x}\right)={-b\pm\sqrt{b^2-4c+8}\over2}\\ x^2+1={-b\pm\sqrt{b^2-4c+8}\over2}x \end{align}$$

Then apply quadratic again for the roots