If b is greater than 1, which of the following must be negative?

If b > 1, then, rearranging that inequality, 1 - b < 0, and b - 1 > 0, and clearly b > 0. So we know the signs of most of the algebraic units in the answer choices.

We have no idea if 2 - b is positive or negative (it can be positive if b is between 1 and 2, and it's zero or negative otherwise) though, so we can rule out the two answers A and D with "2 - b" in them, because those answers will only sometimes be negative, depending on the sign of 2-b. Answer C is a square, so it can't be negative. In answer B we're dividing the positive number b - 1 by the positive number 3b, so that will be positive. That leaves E, which must be negative, because the denominator is positive, while the numerator is negative (if b > 1, then b^2 > 1, and 1 - b^2 < 0, or you can factor the numerator using the difference of squares).

The one issue with plugging in a number here is that if you plug in, say, b = 3, you'll find that answer A and answer E are both negative. You'd only rule out answer A by plugging in a number like b = 1.5 (or anything else between 1 and 2), and if you don't see that you need to do that, you might end up guessing between those two choices.