Pick numbers:

A.) 3x/2. Let's say x=4, then 3x/2 would be even so no.

B.) 3x/2+1. Let's say x=2, then 3x/2+1 would be even so no.

C.) 3x^2. Let's say x=2, then 3x^2 would even so no.

D.) 3x^2/2. Let's say x=2, then 3x^2/2 would be even so no.

The only answer choice left is E.

Another way to solve this:
1) Since Odd* Even = Even, so 3X must be Even.
However, Even/Even=Even or Odd, so we can't decide 3X/2 is even or odd.
For example, 3*2/2=3 (odd), 3*4/2=6(even)
2) Same with the first above. Since 3x/2 could be even or odd, add 1 could yield an even or an odd, so it could come out two answers.
3) even^2 is definitely even. So odd *even= even
4) For this one, since x is even, which means X is divided by 2. Since X will at least have one 2 in it, which means X^2 at least has two 2 in it. Then 3X^2/2 will leave at least one 2 in it. So that means the result is even
5) From the above 4, we know that 3x^2/2 is definitely even, then add 1 will be 100% odd.

So the answer is E.

Not to confuse. but a more abstract efficient way to think about it is any time an even number is divided by 2 it could be odd or even. For instance, 2/2=1 which is odd. and 4/2=2 which is even.

So in order to ensure that an even number we divide by 2 is even, we have to square that number then divide by two. By squaring that number we ensure that there are at least two 2s in numerator as we divide by 2, which will make the result always even. The rest is cake and in this case you will get answer E.

cheers,

sd