i think B is the correct option

Take \(a = -1\) & \(c = 2\) \(b = -3\) \(d = -4\)

Don't forget that this is one of those questions where you can select more than one correct answer (however, in this case, the correct answer is just E)

For the question of this type:
just eliminate the variables with the even powers: a,c because they will not contribute to the answer because they are always going to be positive when the power is applied.
So the remaining b, d variables decide the sign. If they have the positive sign then it is >0 or <0 otherwise.
Be sure they both can be negative also.
SO bd has to be positive for the whole to be positive.

a^8 = always positive, and a itself can be either positive or negative, regardless of other variables' signs
c^6 = always positive, and c itself can be either positive or negative, regardless of other variables' sings
b^7 and d^5 will be positive if the variable within them is positive and negative if the variable is negative. And since we know the total value given is positive, we also know that b and d are either both positive or both negative.

From above:
sign a = +/-
sign c = +/-
sign b = sign d = +/-

So we know any product containing a or c is a tossup and could have either sign, which means that A, B, C, D, and F could all be either positive or negative.

Only E contains b and d, which have to be the same sign. Positive times positive = positive. Negative times negative = positive. So only E must be positive.

Hello from the GRE Prep Club BumpBot!

Thanks to another GRE Prep Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.