The amazing thing is, Galois (biography) was able to demonstrate the "unsolvability" of higher degree polynomials without the benefit of modern algebra. This is like watching the Egyptians build a pyramid with nothing more than human labor and simple machines. The accomplishment is so incredible you can hardly believe it, but there it stands.
Needless to say, I'm not going to build this pyramid by hand. So you need a basic understanding of solvable groups and galois extensions before we proceed. You don't need to know all the theorems in these areas, but you need to understand the definitions.