Answer:
2n + 10 and 4n - 6
Explanation:
To determine whether an expression is even for all integer values of n, we need to check if the expression is divisible by 2 for all integers.
Let's analyze each expression:
1. 2n + 10: This expression can be simplified to 2(n + 5). Since 2 is a factor of 2n for all integers n, and adding 5 does not affect the evenness, this expression is even for all integer values of n.
2. n + 2: This expression represents a linear function with a coefficient of 1. Adding a constant (2) to a linear function does not affect its evenness. Therefore, this expression is not even for all integer values of n.
3. 4n - 6: This expression can be simplified to 2(2n - 3). Since 2 is a factor of 2n for all integers n, and subtracting 3 does not affect the evenness, this expression is even for all integer values of n.
4. 3n + 12: This expression can be simplified to 3(n + 4). Since 3 is not a factor of n for all integers n, this expression is not even for all integer values of n.
5. 2n + 5: This expression represents a linear function with a coefficient of 2. Multiplying a linear function by a non-even number does not preserve its evenness. Therefore, this expression is not even for all integer values of n.
Expressions that are even for all integer values of n: 2n + 10 and 4n - 6.