In order to solve $ \color{blue}{ 4x^{5}+5x^{4}+24x^{3}-36x^{2}+48x = 0 } $, first we need to factor our $ x $.

$ x = 0 $ is a root of multiplicity $ 1 $.

The remaining roots can be found by solving equation $ 4x^{4}+5x^{3}+24x^{2}-36x+48 = 0$.

This polynomial has no rational roots that can be found using Rational Root Test.

Roots were found using quartic formulas