Extending to closed space

Let $\displaystyle X$ be a normed vector space and $\displaystyle M\subseteq X$ a closed subspace. Show that $\displaystyle M+\mathbb{C}x$ is also closed.

I was told to try using the Hahn-Banach theorem. So I know that there exists a function $\displaystyle f\in X^{*}$ such that $\displaystyle \parallel f\parallel =1$ and $\displaystyle f(x)=\delta$. Where $\displaystyle x\in X\cap M^c$ and $\displaystyle \delta=\inf_{y\in M}\parallel x-y\parallel$

However, I don't see how to use this at all.