The problem asks us to expand the expression $(x - y - 1)(x - y + 3)$.

1. Problem Description

The problem asks us to expand the expression $(x - y - 1)(x - y + 3)$.

2. Solution Steps

Let $A = x - y$. Then the expression becomes $(A - 1)(A + 3)$.

Expanding the expression:

$(A - 1)(A + 3) = A^2 + 3A - A - 3 = A^2 + 2A - 3$.

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

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

$(x-y-1)(x-y+3) = (x-y)^2 + 3(x-y) - (x-y) - 3$

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

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

3. Final Answer

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