I suppose you could solve by looking at the answer choices. If the answer is going to be correct for absolutely every value of b, it certainly must be correct when b=1. So we can let b=1 and just plug each answer choice into

x^2 + 2x + 4

to see which gives us the least value of this function. Notice answer A is completely undefined (we get a negative under the root), so cannot possibly be right. If we plug in -2 or 0, the value of the function is 4 in both cases. If we plug in answer D, which is -b^2 = -1, the function's value is 3, which is the smallest value so far. Finally if we plug in answer E, which would be equal to b - 4 = 1 - 4 = -3, the function's value is 6. So the correct answer is -b^2.

But if the GMAT were to ask a question like this, there would always be a way to correctly solve it 'from start to finish', without looking at the answer choices. And I don't see a way to do that here using normal GMAT math. I've never once needed to 'complete the square' in any official GMAT question I've ever solved, nor have I ever needed the quadratic formula, or needed to know how to find the minimum value of a general parabola. I've solved probably close to 10,000 official questions by now, so those techniques just aren't ever required on the test. This question is one that would normally be answered using calculus anyway (and is quite easy if you know calculus), so it's a question I'd bet GMAC would consider unfair, because people with certain educational backgrounds would have a big advantage answering it, and the GMAT is not supposed to be a test of whether you've ever taken a calculus class. It's a test of how well you can reason using only the most elementary facts of mathematics.

So unless I'm not seeing a solution here that uses normal GMAT math, there's really no reason to worry about this question.

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