TITLE: Finding the roots $x^4-4x^3-x^2-8x+4=0$ (contest math)
QUESTION [7 upvotes]: So the problem is :
$x^4-4x^3-x^2-8x+4=0$, find all solutions
A tip that I have gotten, is to divide both sides by $x^2$. I've tried so, but I do not manage to see any further. Do anyone know how this tip could help me?
(Yes, I'm aware that the polynomial above can be factorized into two degree 2 polynomials, which promptly gives me the answer. But that factorization would be extremely hard to spot, which is why I'm asking about the dividing)
Thanks in advance :)
Edit: meant to write $x^2$, not $2$

REPLY [4 votes]: $ x^4 - 4x^3 - x^2 -8 x + 4$ to try to factor it as a product of two  polynomials with degree two I will try this 
$ x^4 -4x^3 -x^2-8x+4=(x^2+ax+c)(x^2+dx+e) $ but the constant term is 4 so we have two choices $ c=1, e=4$ or $ c=2, e=2$ if you choose the second you get the equations $ a+d=-4, 4+ad=-1, 2a+2d=-8$ if you solve them you come up with a solution or maybe there is not a solution.