I am doing a past paper for a first course in algebraic topology. The question is
Calculate the homology groups with $\mathbb Z,\mathbb Z_2,\mathbb Q$ coefficients for $\mathbb RP^4$ and $\mathbb CP^2$ and their connected sum.
I am stuck at trying to calculate the homology of $\mathbb RP^4\#\mathbb CP^2$. I calculated the homology groups of the individual spaces via the standard CW complexes, which are nice and easy. Is there a way of forming a CW complex for the connected sum? Or do I need to use a homology trick for getting at the homology of the connected sum?