If x is real, the maximum and minimum values of expression `(x^(2)+14x+9)/(x^(2)+2x+3)` will be

Options

A. `4,-5`
B. `5,-4`
C. `-4,5`
D. `-4,-5`

Answer

To find the maximum and minimum values of the expression \( y = \frac{x^2 + 14x + 9}{x^2 + 2x + 3} \), we will follow these steps:

### Step 1: Rewrite the expression
Let \( y = \frac{x^2 + 14x + 9}{x^2 + 2x + 3} \).

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ y(x^2 + 2x + 3) = x^2 + 14x + 9 \]
This simplifies to:
\[ yx^2 + 2yx + 3y = x^2 + 14x + 9 \]

### Step 3: Rearrange the equation
Rearranging the equation leads to:
\[ (y - 1)x^2 + (2y - 14)x + (3y - 9) = 0 \]

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

### Step 5: Expand the discriminant
Expanding the discriminant:
\[ D = (2y - 14)^2 - 4(y - 1)(3y - 9) \]
Calculating \( (2y - 14)^2 \):
\[ = 4y^2 - 56y + 196 \]
Calculating \( 4(y - 1)(3y - 9) \):
\[ = 4(3y^2 - 9y - 3y + 9) = 12y^2 - 48y + 36 \]
Thus, we have:
\[ D = 4y^2 - 56y + 196 - (12y^2 - 48y + 36) \]
\[ = 4y^2 - 56y + 196 - 12y^2 + 48y - 36 \]
\[ = -8y^2 - 8y + 160 \]

### Step 6: Set the discriminant greater than or equal to zero
Setting the discriminant \( D \) greater than or equal to zero:
\[ -8y^2 - 8y + 160 \geq 0 \]
Dividing the whole inequality by -8 (which reverses the inequality):
\[ y^2 + y - 20 \leq 0 \]

### Step 7: Factor the quadratic
Factoring the quadratic:
\[ (y - 4)(y + 5) \leq 0 \]

### Step 8: Determine the intervals
The critical points are \( y = -5 \) and \( y = 4 \). The solution to the inequality \( (y - 4)(y + 5) \leq 0 \) gives us the intervals:
\[ -5 \leq y \leq 4 \]

### Conclusion
Thus, the minimum value of \( y \) is \( -5 \) and the maximum value of \( y \) is \( 4 \).