I'm pretty sure that more than half the times I've seen "quintic" are in one single context: talking about the fact that while you can solve polynomial equation

I'm pretty sure that more than half the times I've seen "quintic" are in one single context: talking about the fact that while you can solve polynomial equations of degree 1-4 by doing arithmetic and taking n'th roots, this stops being possible once you get as far as quintic equations. (The "insolubility of the quintic".)