Let
y = x2 + 2x - 11/x - 3
Writing this as a quadratic equation in x, we have
x2 + x (2 - y) + 3y - 11 = 0 ...(1)
The values of x and y are related by this equation and for each value of y, there is a value of x which is a root of this quadratic equation. In order to that this x (or root) is real, the discriminant ≥ 0.
(2 - y)2 - 4(3y - 11) ≥ 0
⇒ y2 - 16y + 48 ≥ 0
⇒ (y - 4)(y - 12) ≥ 0
⇒ y ≤ 4 or y ≥ 12
Hence, y (or the given expression) does not take any value between 4 and 12.
