To factor the expression $x^2 - y^2 - x - y$, let's break it down step-by-step:

1. Group terms: Notice that we can group terms in a way that will make factoring easier: $x^2 - y^2 - x - y = (x^2 - y^2) - (x + y)$

2. Apply the difference of squares: Recognize that $x^2 - y^2$ is a difference of squares, which factors as $(x + y)(x - y)$: $(x + y)(x - y) - (x + y)$

3. Factor by grouping: Now, we see that both terms have a common factor of $x + y$: $= (x + y)(x - y - 1)$

Thus, the factored form of $x^2 - y^2 - x - y$ is: $(x + y)(x - y - 1)$

Would you like a deeper breakdown of each step or have any questions?

Here are five related questions that expand on this problem:

1. How can you apply factoring by grouping to more complex expressions?
2. What other types of identities, like the difference of squares, are useful in factoring?
3. Can you factor $x^2 + y^2 - x - y$ in a similar way?
4. How does recognizing common factors aid in simplifying algebraic expressions?
5. How does factoring help in solving algebraic equations?

Tip: Always look for common factors and known identities (like difference of squares) when factoring complex expressions—it can simplify your work significantly!