Which is the correct simplified form of ( (x+y+z)^2-(x-y+z)^2 )?

Correct answer: (4y(x+z))

The expression contains two squares that differ only in the sign of the middle term. Group the terms that are unchanged by writing the expression as \(A+y)^2-(A-y)^2\), where \(A=x+z\). The identity \( (A+B)^2-(A-B)^2=4AB\) says that this difference is four times the product of the common part and the changing term. This avoids expanding every term separately.

Substituting \(A=x+z\) and \(B=y\) gives \(4(x+z)y=4y(x+z)\). Thus the correct simplified form is option A. Option B is exactly half the required value, option C keeps only an incomplete product, and option D does not represent the difference of these two squares. The grouping and identity also provide a quick check on the result.