Insolubility of the Quintic

At some point in studying Algebra, we learn about the Quadratic Equation: given a polynomial equation of the form

$$a{x}^2+bx+c=0,$$

mathematicians derived a closed-form solution, instantly giving us the two solutions:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

In Medieval Italy, mathematicians further developed a cubic equation, giving us the three solutions to

$$a{x}^3+b{x}^2+cx+d=0,$$

and then the quartic equation, giving us the four solutions to

$$a{x}^4+b{x}^3+c{x}^2+dx+e=0.$$

But then they seemed stumped after this conquest of fourth-degree polynomial equations. There seemed to be something qualitatively different about fifth-degree, or quintic polynomial equations.

Long story short, a 20-year-old named Evariste Galois frantically wrote down a proof while languishing in prison for his participation in the July Revolution,