Topology Winding Number Vector Field Question

Could someone please clarify an answer for me to the following question:

Calculate the winding number of the vectorfield:

[x^3 + 3x(y^2) , 3x(y^2) - (y^3)]

along the curve x^2 + y^2 = 1

I have tried sketching it out for the the +-1 coordinates of x and y and for the diagonals where (x,y) = (root(0.5),root(0.5), however I cannot tell if it is retracing or making complete loops. What is the easiest way to do this?

Could someone please help I would greatly appreciate it!