To understand why the correct answer is option 'B', let's analyze the given quadratic equation x^2 + bx + 1 = 0 and its roots.

The quadratic equation is in the standard form ax^2 + bx + c = 0, where a, b, and c are coefficients.

The discriminant (D) of a quadratic equation is given by the formula D = b^2 - 4ac. It helps determine the nature of the roots.

In this case, the discriminant is b^2 - 4(1)(1) = b^2 - 4.

If the discriminant is less than zero (D < 0),="" then="" the="" quadratic="" equation="" does="" not="" possess="" real="" roots.="" let's="" consider="" the="" cases="" for="" the="" options="" />

a) 3 - b - 3b = 3 - 4b
b) 2 - b - 2 = -b
c) b - 2
d) b - 2

We need to find the option that satisfies the condition D < 0,="" which="" means="" the="" discriminant="" must="" be="" />

Let's substitute each option into the discriminant formula and determine when it is less than zero:

a) D = (3 - 4b)^2 - 4
Expanding and simplifying: D = 9 - 24b + 16b^2 - 4
Simplifying further: D = 16b^2 - 24b + 5

b) D = (-b)^2 - 4
Simplifying: D = b^2 - 4

c) D = (b - 2)^2 - 4
Expanding and simplifying: D = b^2 - 4b + 4 - 4
Simplifying further: D = b^2 - 4b

d) D = (b - 2)^2 - 4
Expanding and simplifying: D = b^2 - 4b + 4 - 4
Simplifying further: D = b^2 - 4b

Now, we need to determine which option satisfies the condition D < />

For options c) and d), D = b^2 - 4b, which is the same as the discriminant of the original quadratic equation. Since the original equation does not possess real roots, options c) and d) will also not have real roots.

For option a), D = 16b^2 - 24b + 5. This is a quadratic equation in terms of b. We can analyze its discriminant separately to determine when it is less than zero.

The discriminant of 16b^2 - 24b + 5 is given by D = (-24)^2 - 4(16)(5) = 576 - 320 = 256.

Since the discriminant is positive, option a) does not satisfy the condition D < 0="" and="" is="" />

Therefore, option b) is the correct answer as the discriminant of b^2 - 4 is always negative, indicating that the quadratic equation x^2 + bx + 1 = 0 does not possess real roots.