Feel free to skip to 10:28 to see how to develop Vladimir Arnold's amazingly beautiful argument for the non-existence of a general algebraic formula for solving quintic equations! This result, known as the Abel-Ruffini theorem, is usually proved by Galois theory, which is hard and not very intuitive. But this approach uses little more than some basic properties of complex numbers. (PS: I forgot to mention Abel's original approach, which is a bit grim, and gives very little intuition at all!)
00:00 Introduction
01:58 Complex Number Refresher
04:11 Fundamental Theorem of Algebra (Proof)
10:28 The Symmetry of Solutions to Polynomials
22:47 Why Roots Aren't Enough
28:29 Why Nested Roots Aren't Enough
37:01 Onto The Quintic
41:03 Conclusion
Paper mentioned: https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf
Video mentioned: https://youtu.be/RhpVSV6iCko