Question 733248
(Assuming you mean the second term is 2x^3)
{{{x^4+2x^3-3x^2-4x+4=0}}}, ___________ 2 sign changes.  Expect 2 positive roots.


Let x become -x.
{{{x^4+2(-x)^3-3x^2-4(-x)+4=x^4-2x^3-3x^2+4x+4}}},_________ 2 sign changes.  Expect 2 negative roots.


Choices for checking roots would be -1, -2, -4, +1, +2, +4.


Use of synthetic division shows -2 root, +2 NOT root, -1 NOT root, +1 root.
So far, Real roots seem to be  -2 and +1.  


The polynomial now rendered to check is {{{x^2+x-2}}}.  Use of general solution to quadratic equation (even though this one is factorable) indicates roots -2 and +1.  They occur in the factorization for the originally given degree 4 polynomial TWICE.


Roots are all REAL numbers, and are -2, +1, -2 (again), and +1 (again).  
{{{highlight(x^4+2^3-3x^2-4x+4=(x+2)^2*(x-1)^2)}}}.