a, b, c, and d are positive integers. If (a + b)(c d) = r, where

I'm happy to help with this.

Statement #1: (a + b) (c + d) = r^2
The prompt told us that (a + b) (c – d) = r. If we divide the statement #1 equation by the prompt equation, we get [(c + d)]/[(c - d)] = r, which is intriguing, but which doesn't, by itself, tell us anything about whether (c + d), the numerator, is a perfect square. This is insufficient.

Statement #2: (a + b) = x^4 y^6 z^2, where x, y, and z are distinct prime numbers
This one definitively tells us that (a + b) is a perfect square, so we know a perfect square times (c - d) equals r, but we know nothing at all about (c + d). This one by itself, doesn't tell us anything. This is insufficient.

Now, consider the statements combined.
Statement #1: (a + b) (c + d) = r^2
Statement #2: (a + b) = x^4 y^6 z^2, where x, y, and z are distinct prime numbers
Now, we know that (a + b) is a perfect square, because it has even powers of all prime factors. We also know that r^2 is a perfect square, so it must also have even powers of all prime factors. This can only mean that (c + d) has even powers of all prime factors, and therefore must be a perfect square.
Another way to say that --- we could re-arrange the statement #1 equation to (c + d) = [r^2]/[(a + b))], and we know this quotient is a positive integer. If the ratio of two squares is an integer, that integer must also be a perfect square.
Either way, the combination of statements is now sufficient to give a definitive answer to the prompt.
Answer = C

One thing that's a little unusual about this question --- with the information in both statements, we could answer the prompt question, but as it turned out, the equation given in the prompt was irrelevant. I don't know that this would happen on the GMAT.

Let me know if anyone reading this has any questions.

Mike