Final Answer:
I) (x / (x + 3) + (x + 2) / (x + 5)) is not closed under addition.
II) (x + 4) / (x^2 + 5x + 6) * (x + 3) / (x^2 - 16) simplifies to (x + 4) / ((x + 2) * (x + 3)).
Step-by-step explanation:
For the first expression, combining the terms ((x / (x + 3)) + ((x + 2) / (x + 5))) results in the inability to find a common denominator for addition. Therefore, the expression is not closed under addition because a common denominator cannot be established, making it impossible to further simplify or combine the terms.
Moving to the second expression, (x + 4) / (x^2 + 5x + 6) * (x + 3) / (x^2 - 16), factorizing the denominators reveals that x^2 + 5x + 6 can be factored to (x + 2) * (x + 3) and x^2 - 16 to (x + 4) * (x - 4). Canceling out the similar terms (x + 3) and (x + 4) leaves us with the simplified expression (x + 4) / ((x + 2) * (x + 3)).
In the first scenario, closure isn't achieved due to the inability to combine the terms under a common denominator. However, in the second scenario, closure is established after simplification, resulting in the expression (x + 4) / ((x + 2) * (x + 3)) where the operation maintains closure as it remains in the same algebraic form without the presence of undefined or unsimplified terms