Let $X$ be $\mathbb{R}\mathbb{P}^2$ with the interior of an embedded disc removed, and let $Y$ be the connected sum of two tori with the interiors of two embedded discs removed. What is $X\#Y$?
(Using the polygonal representation, for $\mathbb{T}\#\mathbb{T}\#\mathbb{R}\mathbb{P}^2$ we have the word $aba^{-1}b^{-1}cdc^{-1}d^{-1}ee$, but how are the removed discs represented? And how do I ultimately determine the resulting space?)