Which of the following is **always** true for all real numbers $x$ and $y$?

Apply the identity $(a + b)^{2} = a^{2} + 2ab + b^{2}$ — one of the seven standard algebraic identities. Option B is exactly this. Why the others fail: - **A** is the most common student error. The cross-term $2xy$ is missing. Counter-example: $(1 + 1)^{2} = 4$, but $1^{2} + 1^{2} = 2$. - **C** is wrong; the correct simplification of $\sqrt{x^{2} + y^{2}}$ does **not** distribute over the sum. (E.g., $\sqrt{3^{2} + 4^{2}} = 5$, not $3 + 4 = 7$.) - **D** confuses $(xy)^{2} = x^{2}y^{2}$ with a sum. The missing $2xy$ in option A is the load-bearing detail this question is designed to catch.