A question from Introduction to topological manifold exercise 5-4

Exe5.4: Suppose M is a compact, connected 2-manifold that contains a subset $B\subset M$ that is homeomorphic to the Möbius band. Show that there is a compact 2-manifold M' such that M is homeomorphic to a connected sum $M'$#$P^2$.

I know if M is a compact, connected 2-manifold, then M has a very good surface presentation. But I have no geometric intuition on the question and can't figure out how to rigorously apply the surface presentation.

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    $\begingroup$ Hint: $\mathbb{RP}^2$ is given by gluing a unit ball along the boundary of a mobius band (see here). $\endgroup$ Jun 2 at 16:13

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