For n being an odd integer which one of the following numbers is an even…

Parity rules: the product of two odd numbers is odd; the product of any integer with an even number is even; the sum of two odd numbers is even; and the sum of an odd and an even number is odd. Since n is odd, n2 = n × n is also odd (odd × odd = odd).

1. 3n − 8: 3n is odd × odd = odd; odd − even (8) = odd, so this expression is always odd.

2. 5n2 + 4: n2 is odd, so 5n2 = odd × odd = odd; odd + even (4) = odd, so this expression is always odd.

3. 3n2 + 5: n2 is odd, so 3n2 = odd × odd = odd; odd + odd (5) = even, so this expression is always even.

4. 4n2 + 5: 4n2 is even regardless of the parity of n2, because any integer multiplied by an even number (4) is even; even + odd (5) = odd, so this expression is always odd.

Cross-check with a concrete odd value: for n = 1, 3n2 + 5 = 3(1) + 5 = 8 (even); for n = 3, 3n2 + 5 = 3(9) + 5 = 32 (even). Every other expression above yields an odd value for these same n.

Therefore, 3n2 + 5 is the expression that is always even when n is an odd integer.