Show that the complex polynomial given by $z^4+2z^2-z+1$ has exactly one root in each quadrant.
I know by the fundamental theorem of Algebra, that the polynomial has exactly four roots.
Now, to show it has exactly one root in each quadrant, it suffices to show that there are two non-real roots in the open right and left half planes, since zeros of polynomials come in conjugate pairs, and they're just the reflection about the real axis in the complex plane.
I think I have to apply Rouché's theorem and compare $z^4+2z^2-z+1$ to a nice function, but I am not immediately sure what that function may be.
Any help would be much appreciated. Thanks in advance!