We start with the given equation:

$(x + y)^2 - (x - y) - 2 = 0$

Step 1: Expand the Square

Expanding $(x + y)^2$:

$x^2 + 2xy + y^2$

Thus, the equation becomes:

$x^2 + 2xy + y^2 - (x - y) - 2 = 0$

Step 2: Rearrange the Terms

Rewriting the equation:

$x^2 + y^2 + 2xy - x + y - 2 = 0$

We now seek to factorize it.

Step 3: Try Factoring

We attempt factoring by considering expressions that multiply to $x^2 + y^2 + 2xy - x + y - 2$. Checking for a binomial factorization of the form $(x + y + a)(x + y + b)$, we set:

$(x + y + a)(x + y + b) = x^2 + 2xy + y^2 + (a + b)(x + y) + ab$

Matching with $x^2 + 2xy + y^2 - x + y - 2$, we get:

$a + b = -1, \quad ab = -2$

Solving for $a$ and $b$, the possible values are $a = -2, b = 1$, so the factorization is:

$(x + y - 2)(x + y + 1) = 0$

Final Answer:

$(x + y + 1)(x + y - 2) = 0$

Would you like further details or have any questions?

Related Questions:

1. How do you factor quadratic expressions step by step?
2. What is the geometric interpretation of $(x + y + 1)(x + y - 2) = 0$?
3. How do you solve quadratic equations by factoring?
4. Can you show another example of factoring a quadratic expression?
5. What are the different methods of solving quadratic equations?

Tip: Always check your factorization by expanding it back to the original equation!