  This is an English translation of Felix Klein's classical paper "\"Uber die
Aufl\"osung der allgemeinen Gleichungen f\"unften und sechsten Grades (Auszug
aus einem Schreiben an Herrn K. Hensel)" from 1905 and is put in modern
notation. The original work first appeared in the Journal for Pure and Applied
Mathematics (Volume 129) and then was reprinted in Mathematische Annalen
(Volume 61, Issue 1).
  Klein's work (including his "Lectures on the Icosahedron and the Solution of
Equations of Fifth Degree") lies at the heart of the 19th and 20th work on
solving generic polynomials. In this paper, Klein summarizes his approach to
solving the generic quintic and sextic polynomials. He also lays the foundation
for the modern framework of resolvent degree.
