The correct answer is option 4: 1 + x + y < (1 + x + y).

Let`s analyze each option to understand why this is the correct answer for positive real numbers x and y:

- Option 1: xy > x + y

This inequality is not always true for positive real numbers. It could be true for certain values of x and y, but it is not always true. Therefore, this option is incorrect.

- Option 2: (x + y) < (x + y)^2

This inequality is not always true for positive real numbers. It is equivalent to saying that (x + y) is less than its square, which is not always true. Therefore, this option is incorrect.

- Option 3: x + y < x^2 + y^2

This inequality is also not always true for positive real numbers. It could be true for certain values of x and y, but it is not always true. Therefore, this option is incorrect.

- Option 4: 1 + x + y < (1 + x + y)

This inequality is always true for positive real numbers because any positive number added to itself will always be greater than the original number. Therefore, this