How to solve an nth degree polynomial equation

There is no perfect answer to this question. For polynomials up to degree 4, there are explicit solution formulas similar to that for the quadratic equation (the Cardano formulas for third-degree equations, see here, and the Ferrari formula for degree 4, see here).

For higher degrees, no general formula exists (or more precisely, no formula in terms of addition, subtraction, multiplication, division, arbitrary constants and $n$-th roots). This result is proved in Galois theory and is known as the Abel-Ruffini theorem. Edit: Note that for some special cases (e.g., $x^n - a$), solution formulas exist, but they do not generalize to all polynomials. In fact, it is known that only a very small part of polynomials of degree $\ge 5$ admit a solution formula using the operations listed above.

Nevertheless, finding solutions to polynomial formulas is quite easy using numerical methods, e.g., Newton's method. These methods are independent of the degree of the polynomial.

I'd like to show you this flowchart which summarizes all methods of solving up to quartic polynomials manually:

Abel's impossibility theorem states that there is no algebraic solution to polynomial equations of degree five or higher

But Jordan has shown that any algebraic equation can be solved using modular functions. There are explicit formulas without the need to use Tschirnhausen or other transformations. However, application of this theorem in practice is very difficult because of the complexity of the relevant hyperelliptic integrals and higher genus theta functions. (General formulas see here)

There are general formulas for the general equations $$x^n-x+t=0$$ and $$ax^{2\mu}+bx^\mu-x^\nu+c=0$$ (see here and here)

If I understand the question correctly: there is no general expression for finding roots of polynomials of degree 5 or more. See here

For degrees 3 and 4 the Wikipedia entries are quite good.