If f(x) = x²/b² + 2x + 4, then for each non-zero b f(x) is the least when x equals
A. -1 – √(1 – 4b)
B. -2
C. 0
D. -b²
E. b – 4

(D) x²/b² + 2x + 4 = (x/b)² + 2(b/b)x + b² + 4 – b² = (x/b + b)² + (4 – b²).
Any square is a non-negative number, so f(x) is the least, when (x/b + b)² = 0.
x/b + b = 0
x = -b²
The correct answer is D.
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Does bf(x) not mean
b(x²/b² + 2x +4) = bx²/b² + 2b/x + 4b?

I understand what you've done by multiply and divide by b and add and subtract b² but this is not what I understand the question. Or is there any mistype in this question? Please clarify. thanks

Apparently, you have overlooked the words "for each" and the space in between "b" and "f(x)" in the question statement. In this particular question we do NOT deal with bf(x), but only with f(x), while the parameter b can possess any value, except 0.

This is one of the toughest questions indeed. It's main idea is to use the well-known formula (a + c)² = a² + 2ac + c² backwards. But you must be careful with identifying what a and c stand for in this particular question. You should start with a (a stands for x/b) and then identify c from the term with the variable (2x). It stands for 2ac.
The hint for using this well-known formula comes from the look of the expression of the function itself. If you get a very good understanding of how this formula is applied in this particular case and why this quadratic function possesses the least value when that square is 0, you'll improve your timing and eliminate a bunch of possible mistakes when dealing with easier cases and other adjoining types of questions.

P.S. If you studied properties of the quadratic equation and can use the formula for finding the required value of the variable, go for it. If you didn't - the provided explanation shows how to solve this question using just the well-known formula ((a + c)² = a² + 2ac + c²), and the property of the square (≥ 0).