Question 527290
You're correct. All you need to do now is factor it to 4(k^2 + 2k + 8). Since k^2  + 2k + 8 is an integer, the expression is divisible by 4.


Another way to prove it is to use modular arithmetic, or arithmetic dealing with remainders. If m is an odd integer, then m ≡ 1 mod 2 (i.e. m divided by 2 leaves a remainder 1. The ≡ sign means "equivalent"). Then m^2 ≡ 1 (mod) 4, 2m ≡ 2 (mod 4), and 5 ≡ 1 (mod 4). We can add modulos or residues the same way we add numbers; in this case, m^2 + 2m + 5 ≡ 1 + 2 + 1 ≡ 4 ≡ 0 (mod 4). Note that if a number is m mod n, we can subtract any multiple of n and still obtain an equivalent result. Since the expression is 0 mod 4, it is divisible by 4 (since it leaves remainder 0).