Vyshak's solutions are great.
Let's examine his second approach.

If -2 < x < 5, then it's possible that x = 5, in which case x² = 25
In other words, it's possible that x² = 25
When we check the answer choices, we see that...
Answer choice A says that x² < 16. In other words, it says that x² cannot equal 25. ELIMINATE A
Answer choice C says that x² < 16. In other words, it says that x² cannot equal 25. ELIMINATE C

Also, if -2 < x < 5, then it's possible that x = 0, in which case x² = 0
In other words, it's possible that x² = 0
When we check the remaining answer choices, we see that...
Answer choice B says that 4 < x². In other words, it says that x² cannot equal 0. ELIMINATE B
Answer choice D says that 0 < x². In other words, it says that x² cannot equal 0. ELIMINATE D

We're left with 1 answer choice...E

Cheers,
Brent
_________________

Square of any number can never b negative, answer D is correct.

From the stem x CAN be 0, and if it is, then D is wrong. The correct answer is E. Please re-read the solutions

D option does not say equal to zero, it says greater than zero, if you see option D, there is not a sign of equal to

This to me is a simple question that is complicated by GMAT trickery in the way the answer choices are presented. the smallest value for x^2 is 0, and 0 is always greater than -4, so option E still fulfills the demands of the question. It is always true that x^2 is greater than or equal to -4 because it is always greater than -4.

Again it comes down to the way the question is presented: "which of the following MUST be true". In theory, answer E could have been -1,000,000 ≤ x^2 < 1,000,000 and it would still be the only correct answer because it is the only solution offered that contains the actual possible values of x^2. notice the maximum value of x^2 is 25 but answer E allows for up to 36... we know x^2 cannot get that big, but the way the solution is presented means x^2 never actually needs to reach ~36 in order for it to still satisfy those constraints. This is the same idea on the negative lower bound of x^2.

Also, x^2 = 0 is a possible solution if x=0.

I think you don't understand the logic of the question. The question asks: if \(-2 < x \leq {5}\), then which of the following MUST be true? So, we KNOW that \(-2 < x \leq {5}\) and should evaluate which of the options must be correct. If \(-2 < x \leq {5}\), then only E must be true.
_________________

Many students will assume that E is incorrect, because x² CANNOT equal -4. However, answer choice E does not suggest that x² can equal -4, it only states that x² must be GREATER THAN or equal to -4.

Here's a similar example: If Joe has more than 5 shirts, which of the following can we conclude with certainty?
A) Joe owns more than 6 shirts.
NO. We can't conclude this, because it's possible that Joe owns exactly 6 shirts, and 6 is not greater than 6.

B) Joe owns more than -3 shirts.
YES. If Joe owns more than 5 shirts, then he definitely owns more than -3 shirts?
Does this mean that Joe COULD own -2 shirts? No. If just means that we can be certain that he owns more than -3 shirts.

Does that help?

Cheers,
Brent
_________________