How is the general solution for algebraic equations of degree five formulated?

In a book on neural networks I found the statement:

The general solution for algebraic equations of degree five, for example, cannot be formulated using only algebraic functions, yet this can be done if a more general class of functions is allowed as computational primitives.

What are the "more general class of functions"?

You can, for example, define an operation analogous to an $n$th root, except that instead of saying that $x=\sqrt[5] y$ if $x^5-y=0$, you say that $x=BR(y)$ if $x^5 +x -y=0$.

You can then express the solution of the general quintic in terms of $+, -,\times,\div,$ ordinary radicals, and $BR()$.

See Bring radical for more complete details, especially the section on solution of the general quintic.

Felix Klein has a small book called "Lectures on the icosahedron and the solution of equations of the fifth degree", where he develops a method of solving the quintic using modular forms. The relationship between the isocahedron and the general quintic is that the automorphism group of the isocahedron is $S_5$, which is also the Galois group of the general quintic.

Googling has turned up this introductory blog post on the topic, and this expository article.