When it comes to Integer Property questions, we can often answer the question by testing a value that satisfies the given information.

Given: n is an integer and \(n^3\) is an odd integer
So, it could be the case that, n = 1, since \(1^3 = 1\), and 1 is odd.

Now we'll plug n = 1 into each answer choice to see which one yields and even integer.

(A) \(2(1)^2 + 1=3\) ELIMINATE

(B) \(1^4=1\) ELIMINATE

(C) \(1^2 + 1=2\) KEEP

(D) \(1(1 + 2)=3\) ELIMINATE

(E) \(1\) ELIMINATE

Answer: C