Fifth Degree Equation

Dear All

I have come across an article about Galois theory written by Dan Goodman, where in the end, the author gives an example of a polynomial of fifth degree equation that is not solvable by radicals, which is not true
Here is the article web link,

http://nrich.maths.org/public/viewer.php?obj_id=1422

I therefore should like to ask the following question!

Is there a method to decide fast whether the reduced form of quintique equation with integer coefficients is solvable or not

Examples, consider the following equations

x^5 +5*x +4 = 0

x^5 -(214524)*x +969408 = 0

Thanking your comments


Bassam Karzeddin
Al-Hussein Bin Talal University
JORDAN
.