The expression to factorize is:

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

Step-by-Step Factorization:

1. Recognize the common factor: Both terms in the expression contain $(x + y)$. So, factor $(x + y)$ from both terms.

$(x + y)\left[(x + y) + 3\right]$

2. Simplify the expression inside the parentheses:

$(x + y)(x + y + 3)$

Thus, the factorized form of $(x + y)^2 + 3(x + y)$ is:

$(x + y)(x + y + 3)$

Would you like further clarification or details?

Here are some related questions for deeper exploration:

1. How would you factor a cubic expression involving $(x + y)$?
2. What happens if we add a constant term to the original expression? How would you factor it then?
3. Can this method be applied to more complex binomials, such as $(x - y)$ or $(x + 2y)$?
4. How does the distributive property help in factorization?
5. Can you expand the factorized form to verify the original expression?

Tip: Always look for common terms in factorization to simplify expressions early in the process.