How to reduce an equation of nth degree to a quadratic?

Hi everyone, I was studying the cyclotomic equation and I came across an example:

x^4 + x^3 + x^2 + x +1 = 0 ; this equation cannot be factorised neither using Ruffini's rule nor replacing an x with t. The book said that it is sufficient to divide all by x^2 and rearrange the terms to obtain --> x^2 + 1/x^2 + x + 1/x + 1 = 0, then let w be equal to x + 1/x, and the equation (I also have verified) becomes w^2 + w -1 = 0.

My question is: does it exist a formula (or a procedure) that allows to reduce an equation with a degree greater than 2 to a quadratic? Procedures other than replace x^4 with t and x^2 with t.

In fact there would be Abel-Ruffini theorem to take into account...which says that there is not a general algebraic solution for quintic equations and higher degrees...

Could you help me please?