1. **Stating the problem:** We need to determine which expressions are even for all integer values of $n$. An expression is even if it is divisible by 2 for every integer $n$. 2. **Check each expression and factor out terms related to $n$: ** - Expression 1: $2n + 8$ - $2n$ is always even because it's 2 times an integer. - 8 is even. - Sum of even + even = even, so $2n + 8$ is even for all $n$. - Expression 2: $5n + 10$ - $5n$ may be odd or even depending on $n$ (e.g. for $n=1$, $5n=5$ odd). - 10 is even. - Odd + even = odd, so $5n + 10$ is not always even. - Expression 3: $2n + 3$ - $2n$ is even. - 3 is odd. - Even + odd = odd, so not always even. - Expression 4: $n + 2$ - $n$ can be odd or even. - 2 is even. - Sum depends on $n$, so expression is not always even. - Expression 5: $4n - 14$ - $4n$ is even (4 times any integer). - 14 is even. - Even - even = even, so $4n - 14$ is even for all $n$. 3. **Conclusion:** The expressions always even for all integer $n$ are $2n + 8$ and $4n - 14$. **Final answer:** $2n + 8$ and $4n - 14$ are always even for all integers $n$.