Exact Equations & Substitutions · Superchallenge problem

The equation $2x(1 + y)\,dx + (x^2 + 4y)\,dy = 0$ is exact. For $c > 0$, the solution curve through $(c, 0)$ is a closed curve around the origin when $c$ is small, but not for every $c$. Find the supremum of the values of $c$ for which it is a closed curve.