Which of the following represents an even integer for each integer value of n?

1. Translate the problem requirements: We need to find which expression always produces an even integer regardless of what integer value we substitute for n. An even integer means the result is divisible by 2.
2. Test strategic integer values: Check each expression using both even and odd values of n to see if the result is always even, since if an expression works for all integers, it must work for both even and odd cases.
3. Apply even/odd arithmetic rules: Use the fact that even ± even = even, odd ± odd = even, and even × anything = even to analyze the expressions algebraically.
4. Verify the winning expression: Confirm that the identified expression indeed produces even results for all integer inputs by checking its mathematical structure.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for. We need to find which expression will always give us an even integer, no matter what integer we plug in for n.

An even integer is any whole number that can be divided by 2 evenly. Since any integer is either even or odd, if an expression works for both even and odd values of n, then it works for all integers.

Process Skill: TRANSLATE - Converting the problem requirement into a clear mathematical test

2. Test strategic integer values

Let's test each expression using n = 1 (odd) and n = 2 (even):

Choice A: \(\mathrm{n^2 - 3n + 4}\)

n = 1: \(1 - 3 + 4 = 2\) (even ✓)

n = 2: \(4 - 6 + 4 = 2\) (even ✓)

Choice B: \(\mathrm{n^2 + 2n + 4}\)

n = 1: \(1 + 2 + 4 = 7\) (odd ✗) → eliminate B

3. Apply even/odd arithmetic rules

For \(\mathrm{n^2 - 3n + 4}\):

• If n is even: even - even + even = even
• If n is odd: odd - odd + even = even

Process Skill: CONSIDER ALL CASES - Checking both even and odd values systematically

4. Verify the winning expression

Additional checks (n = 3, 0, -1) all yield even results. Algebraically, n(n−3) is always even and adding 4 preserves evenness.

Final Answer

Choice A: \(\mathrm{n^2 - 3n + 4}\)

Common Faltering Points

1. Testing only positive or only one parity. 2. Arithmetic errors when substituting. 3. Misapplying parity rules (e.g., odd−odd). 4. Not verifying "always" requirement with both even and odd cases.

Alternate "Smart Numbers" Solution

Test n = 2 (even) and n = 1 (odd) to eliminate all but A, then confirm with n = 3.