In each of the choices, three integers are multiplied together. For the product of the integers to be divisible by 4, either one of the integers must be divisible by 4, or two of the integers must be divisible by 2. Since we don't know anything about the specific value of y, we can't determine whether any of the integers are divisible by 4.
Consider each choice:
(A) If y is even, then 2y is divisible by 4, so the whole expression is divisible by 4. If y is odd, then y+1 is even and y-1 is even, so the whole expression is divisible by 4.
(B) If y is even, then y is even and 2y + 2 is even, so the expression is divisible by 4. If y is odd, then 2y + 2 must be divisible by 4. 2y + 2 = 2(y + 1), and if y is odd, y + 1 is even. 2 times an even is divisible by four.
(C) If y is even, then 2y - 4 is divisible by 4. 2y - 4 = 2(y - 2), and if y is even, then y - 2 is even. 2 times an even is divisible by 4. If y is odd, both y + 3 and 2y - 4 are even, so the expression is divisible by 4.
(D) If y is even, 2y is divisible by 4. If y is odd, 2y is even (but not divisible by 4), and y + 4 and y - 2 are both odd. Thus, the expression may not be divisible by 4. This looks like our answer.
(E) If y is even, 2y + 4 must be divisible by 4. If y is odd, both y + 1 and y - 3 must be even, so the expression must be divisible by 4.
Choice (D) is correct.