Analytic continuation of a function on the right half-plane to a region enclosing the circle ${ lvert z rvert = 3}$

I am trying to solve the following question:

Show that there exists an analytic function $f$ in the open right half-plane such that $(f(z))^2 + 2f(z) equiv z^2$. Show that your function $f$ can be continued analytically to a region
containing the set ${z in mathbb{C} , lvert z rvert = 3}$.

I can solve the first part. Since $z^2+1$ is a non-vanishing holomorphic function in the open right half-plane, which is simply connected, the function admits an analytic square root $h(z)$ in this region; it then suffices to choose $h(z)-1$ as our required function $f(z)$.

I don’t know how to proceed with the analytic continuation part though. Any suggestions?