Final Answer:
The given expression cannot be solved directly as it involves two distinct operations. Please provide clearer instructions or split the operations for a more precise solution.
Step-by-step explanation:
The problem appears to present two distinct operations: the addition of fractions in the first expression and multiplication of rational expressions in the second. The first expression (x / (x + 3) + (x + 2) / (x + 5)) involves adding fractions. To simplify, a common denominator must be found, yet upon evaluation, there isn't an evident common denominator between (x + 3) and (x + 5) to proceed with addition. Therefore, closure cannot be proven in this case due to the lack of a common denominator for the fractions.
In the second expression, (x + 4) / (x^2 + 5x + 6) * (x + 3) / (x^2 - 16), the task seems to entail multiplying rational expressions. Simplification involves factoring the denominators and canceling out similar terms. Upon factorization, it's discovered that (x^2 + 5x + 6) factors into (x + 2) * (x + 3) and (x^2 - 16) factors into (x + 4) * (x - 4). Canceling similar terms results in (x + 4) / ((x + 2) * (x + 3)). This simplified expression adheres to closure, as it retains the same algebraic form and doesn't possess undefined or unsimplified elements.
However, the instructions provided do not clearly indicate the sequence of operations or the intended outcome. As a result, the closure for the given expression cannot be directly established due to the ambiguity in the operations presented. Clarifying the instructions or breaking down the operations separately would facilitate a more accurate solution.