The fastest approach, in my opinion, is to replace g and h with two convenient odd numbers.
The easiest odd numbers to work with are g = 1 and h = 1.
Plug these values in answer chances choice to get:
A. g/h = 1/1 = 1. Odd. Keep.
B. (g + h)² = (1 + 1)² = 4. EVEN. ELIMINATE.
C. hg + h = (1)(1) + 1 = 2. EVEN. ELIMINATE.
D. (g - h)² =(1 - 1)² = 0. EVEN. ELIMINATE.
E. (gh)³ = ((1)(1))³ = 1³ = 1. Odd. Keep.

At this point, we can quickly show that answer choice A (g/h) need not be an integer.
For example, if g = 1 and h = 3, then g/h = 1/3, which is not an odd number, which means we can eliminate answer choice A.

By the process of elimination, the correct answer is E.