Nice question!!

The given information, \(|x|y > x|y|\), provides some very useful information about x and y.

First recognize that:
i) If x and y are both POSITIVE, then \(|x|y = x|y|\)
ii) If x and y are both NEGATIVE, then \(|x|y = x|y|\)
iii) If x is POSITIVE and y is NEGATIVE, then \(|x|y < x|y|\)
iv) If x is NEGATIVE and y is POSITIVE, then \(|x|y > x|y|\)

You can verify this by testing various values of x and y

Notice that case iv is the only case that satisfies the given inequality \(|x|y > x|y|\)
So, it must be the case that x is NEGATIVE and y is POSITIVE

Now let's move on to the comparison. We have:
Quantity A: \((x+y)^2\)
Quantity B: \((x-y)^2\)

Expand and simplify both quantities to get:
Quantity A: \(x^2 + 2xy + y^2\)
Quantity B: \(x^2 - 2xy + y^2\)

Subtract \(x^2\) and \(y^2\) from both quantities to get:
Quantity A: \(2xy\)
Quantity B: \(-2xy\)

Add \(2xy\) to both quantities to get:
Quantity A: \(4xy\)
Quantity B: \(0\)

Since we already know x is NEGATIVE and y is POSITIVE, then we also know that \(xy\) is NEGATIVE, which also means \(4xy\) is NEGATIVE.

Since 0 > some NEGATIVE value, the correct answer is B

Cheers,
Brent