Solving a Nice Quartic in Two Ways (x^4+x^3+x^2+x+1=0)

Picked one by one from a mixed channel; title passed two strict editor passes and the clickbait rules.

A 9-minute video from SyberMath: “Solving a Nice Quartic in Two Ways (x^4+x^3+x^2+x+1=0)”.

Video by SyberMath, played through YouTube's embedded player.