= Solution
The <image> <contour> has polar representation $2\cos2\theta\,e^{3i\theta}$. It is invariant under complex conjugation and under reflection through the origin. As $\theta$ runs from zero to $2\pi$, the successive real-axis visits are
$$
2,\ 0,\ 1,\ -1,\ 0,\ -2,\ 0,\ -1,\ 1,\ 0,\ 2,
$$
at angles $0,\pi/4,\pi/3,2\pi/3,3\pi/4,\pi,5\pi/4,4\pi/3,5\pi/3,7\pi/4,2\pi$. To specify the crossing direction, differentiate:
$$
\frac d{d\theta}p(e^{i\theta})=i(5p(e^{i\theta})-4e^{i\theta}).
$$
Its imaginary component is $6$ at the point $2$, $-6$ at $-2$, $3$ at either visit to $1$, and $-3$ at either visit to $-1$. At a visit to zero it is $-4\cos\theta$, nonzero for each of the four angles. Therefore all these visits are transverse crossings of the real axis, even though different arcs share the same <image> point.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/ib/paper-4-polynomial-image.png]
{title=Oriented image of the unit circle under z to the fifth power plus z, showing its real-axis crossings}
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