For a start, f(z) has no positive real roots (it has the value 1 at z=0, and thereafter it increases, as you can check by elementary calculus). The polynomial has real coefficients, so its non-real roots occur in complex conjugate pairs. Therefore you can look for the number of roots with positive real part, and divide that number by 2 to get the number in the first quadrant.

So rather than looking at the winding number round a quadrant, I would look for the winding number of f(z) round a D-shaped contour consisting of a semicircle in the right-hand half plane, going from –iR to +iR, followed by a straight line segment down the imaginary axis from +iR to -iR. Here, R can be any number greater than or equal to 4.

To work out the winding number of f(z) as z goes round this contour, I would think in purely geometric terms rather than trying to evaluate the integral of f'/f. As z goes round the semicircle, f(z) is dominated by the term , so it goes 4 times round the origin. For the straight line portion of the contour, write z=iy. Then The real part of this is clearly always positive, so f(iy) cannot encircle the origin at all. Thus the winding number remains at 4, and the number of roots in the first quadrant is 2.