Intuitive reasoning why are quintics unsolvable?

Question

I know that quintics in general are unsolvable, whereas lower-degree equations are solvable and the formal explanation is very hard. I would like to have an intuitive reasoning of why it is so, accessible to a bright high school student, or even why it should be so. I have also read somewhere that any $n$-degree equation can be depressed to the form $ax^n + bx + c$. I would also like to know why or how this happens, at least for lower degree equations. I know that this question might be too broad and difficult, but this is a thing that has troubled me a lot. To give some background, I recently figured out how to solve the cubic and started calculus, but quartics and above elude me. EDIT: It was mentioned in the comments, that not every $n$-degree equation can be depressed to the form $ax^n + bx + c$, although I recall something like this I have read, anyways, I wanted to find out the same for quintics.

Answer 1

You have some excellent questions! Too broad to be answered properly in this chat box. Do some research on the "fundamental theorem of algebra" and it will give you some of your answers.

And yes, all nth-degree polynomials can be decomposed into a product of linear and quadratic terms.