Does the "$\neq$" in, for instance, "Prove that $(x+y)^2\neq x^2+y^2$" mean "never equal"? "not equivalent"? I have the problem:

Prove that $(x + y)^2 \not= x^2 + y^2$

I have answered it with something like this:
$$(x+y)^2 = x^2 + 2xy + y^2$$
$$(x^2 + 2xy + y^2) - (x^2 + y^2) = 2xy$$
$$\therefore (x + y)^2 = x^2 + y^2 \Rightarrow 2xy = 0$$
$$\therefore (x+y)^2 = x^2 + y^2 \Rightarrow x=0 \lor y=0$$
(Apologies if I'm using symbols incorrectly here, I'm very new to this sort of thing, and as I'm teaching myself I have no one to point these sorts of things out to me)
My problem is that, as far as I can tell, this proves that $(x + y)^2 = x^2 + y^2$ for at least some combination of values for $x$ and $y$. So I suppose my question is, for this kind of proof question, am I being asked to prove that the two statements are not equivalent, or that they are never equal?
Here's a picture of the problem:

 A: The statement $(x+y)^2\ne x^2+y^2$ contains "free variables" and whether it is true or not depends on the values of $x$ and $y$.
If $x=y=0$, then this statement is clearly false.
It is correct to say that "there exist real numbers $x$ and $y$ such that $(x+y)^2\ne x^2+y^2$".
A: You are correct. The statement ought to be $$\text{ Prove that } (x+y)^2\not\equiv x^2+y^2$$
For which it suffices to simply give a counterexample: $4=(1+1)^2\neq1^2+1^2=2$.
The technical distinction between $=$ and $\equiv$ is often overlooked, especially in high school courses, but it is very important. $=$ means it works for a specific value or set of values, $\equiv$ means it works for all values.
A: Without further information there is a couple ways to interpret the problem statement.

*

*Perhaps it is to prove that $(x+y)^2 \neq x^2 + y^2$ fails to be an identity. Which is to say there exists values for which it fails. Thus we prove it is not an identity by giving a counter-example to $(x+y)^2\neq x^2 + y^2$.


*We could interpret it to be asking us to prove $(x+y)^2 > x^2 + y^2$ or $(x+y)^2 < x^2 + y^2$. However, this also is case dependent. So again would fail to be an identity.
My guess would be that it is asking for the first interpretation I have listed above.
