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