I would label the 4 integers, from left to right, top to bottom as and so we have:
Now the two factors on the left are the sums of two numbers that are diagonal with respect to one another, and are therefore neither in the same row nor in the same column. The largest possible sum for one of these pairs (let's use the first) is:
If we want both numbers to be as large as possible, then one could be 72 and the other 71. If we want just 1 of the numbers to be as large as possible, then one could be 143 and the other 1.