If `x` is real, then the value of the expression `(x^2+14 x+9)/(x^2+2x+3)` lies between

Options

A. [4, 5]
B. [-4, 5]
C. [-5, 4]
D. none of these

Explanation

To solve the problem, we need to find the range of the expression \(\frac{x^2 + 14x + 9}{x^2 + 2x + 3}\) for real values of \(x\). Let's denote this expression as \(y\).

### Step-by-Step Solution:

1. **Set up the equation**:
\[ y = \frac{x^2 + 14x + 9}{x^2 + 2x + 3} \]

2. **Cross-multiply to eliminate the fraction**:
\[ y(x^2 + 2x + 3) = x^2 + 14x + 9 \]
This simplifies to:
\[ yx^2 + 2yx + 3y = x^2 + 14x + 9 \]

3. **Rearrange the equation**:
\[ (y - 1)x^2 + (2y - 14)x + (3y - 9) = 0 \]

4. **Use the discriminant condition**:
For \(x\) to be real, the discriminant of the quadratic equation must be non-negative:
\[ D = (2y - 14)^2 - 4(y - 1)(3y - 9) \geq 0 \]

5. **Expand the discriminant**:
\[ D = (2y - 14)^2 - 4[(y - 1)(3y - 9)] \]
Expanding both parts:
\[ D = 4y^2 - 56y + 196 - 4[(3y^2 - 9y - 3y + 9)] \]
\[ = 4y^2 - 56y + 196 - 4(3y^2 - 12y + 9) \]
\[ = 4y^2 - 56y + 196 - 12y^2 + 48y - 36 \]
\[ = -8y^2 - 8y + 160 \]

6. **Simplify the discriminant**:
\[ -8(y^2 + y - 20) \geq 0 \]
Dividing by -8 (which reverses the inequality):
\[ y^2 + y - 20 \leq 0 \]

7. **Factor the quadratic**:
\[ (y - 4)(y + 5) \leq 0 \]

8. **Determine the intervals**:
The roots of the equation are \(y = 4\) and \(y = -5\). The quadratic opens upwards, so the solution to the inequality \( (y - 4)(y + 5) \leq 0 \) is:
\[ -5 \leq y \leq 4 \]

9. **Conclusion**:
Therefore, the value of the expression \(\frac{x^2 + 14x + 9}{x^2 + 2x + 3}\) lies between:
\[ y \in [-5, 4] \]