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Which of the following is correct if x is a real number and (x - 11)(x - 3) is negative?
A. x^2 + 5x + 6 < 0
B. x^2 + 5x + 6 > 0
C. 5 - x < 0
D. x - 5 < 0
E. 11 - x > 0

(x-11)(x-3) will be negative when 3<x<11 and;
x^2+5x+6=(x+2)(x+3) and this equation is negative when -3<x<-2 and will be positive when x>-2 or either x<-3
As a result I think none of these choices can be true in this special case, because regarding to the above situations if you check, it can be seen that none of these answers is true!!!

You are right about the solution set. However, you inferred it incorrectly.

As you had said when (x - 11)(x - 3) < 0, 3 < x < 11.

Choice B results in a solution set x > -2 or x < -3.

So, a value of x between 3 and 11 will certainly satisfy the condition that x^2 + 5x + 6 > 0.

Yes, you are right, but as you resulted in the selection of choice (B), also we can say that the choice (E), because when 3<x<11 then it can satisfy choice (E) which is 11-x>0. Am I right? I just followed your logic resulted in the (B) selection and with similar logic choice (E) can be true also!

You are absolutely right.

Error regretted. Both B and E are correct answers.