We need to expand the expression $(x - y + 3)(x - y - 3)$.

1. Problem Description

We need to expand the expression $(x - y + 3)(x - y - 3)$.

2. Solution Steps

Let $A = x - y$. Then the expression becomes $(A + 3)(A - 3)$.

This is a difference of squares:

$(a + b)(a - b) = a^2 - b^2$.

So, $(A + 3)(A - 3) = A^2 - 3^2 = A^2 - 9$.

Substituting back $A = x - y$, we have:

$(x - y)^2 - 9$.

Now we expand $(x - y)^2$ using the formula:

$(a - b)^2 = a^2 - 2ab + b^2$.

So, $(x - y)^2 = x^2 - 2xy + y^2$.

Therefore, the expression becomes:

$x^2 - 2xy + y^2 - 9$.

3. Final Answer

$x^2 - 2xy + y^2 - 9$