Expand the expression $(x - y - 1)(x - y + 3)$.

1. Problem Description

Expand the expression $(x - y - 1)(x - y + 3)$.

2. Solution Steps

Let $A = x - y$. Then the given expression becomes:

$(A - 1)(A + 3)$.

Expanding this expression, we get:

$A^2 + 3A - A - 3 = A^2 + 2A - 3$.

Substitute $A = x - y$ back into the expression:

$(x - y)^2 + 2(x - y) - 3$.

Now expand $(x - y)^2$:

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

Substitute this back into the expression:

$x^2 - 2xy + y^2 + 2(x - y) - 3$.

Distribute the 2:

$x^2 - 2xy + y^2 + 2x - 2y - 3$.

3. Final Answer

$x^2 - 2xy + y^2 + 2x - 2y - 3$