Just fill in a table

There are only eight possible out comes, so write vertically 1 and 2 for x

and 1,2,3,4 horizontally for y.

Now fill in the eight positions and go from there

Next, obtain the marginals by summing across and down.

The event of X+Y=4, occurs when X=2 and Y=2 OR X=1 and Y=3

Just add the probabilities of those possible outcomes.