As one can "see", the polynomial is $z^4+1+z(z-1)^3$. On the real axis $z=x$, outside the interval $[0,1]$, all terms are positive. Inside that interval the last term is smaller than $1$ in absolute value, so that the argument of the function value stays at the angle $0$.
On the imaginary axis $z=iy$ the polynomial value can be decomposed into real and imaginary parts to get
The quadrants that are traversed are
at no point is the real negative axis crossed, giving a winding number of $0$ along the half axes.
Thus the winding number for each quadrant is determined by the leading term $2z^4$ along large quarter circles which in the end means that there is exactly one root inside every quadrant.