Find the range of rational expression `y =(x^(2) -x+4)/(x^(2) +x+4)` is x is real.

Answer

To find the range of the rational expression \( y = \frac{x^2 - x + 4}{x^2 + x + 4} \) for real values of \( x \), we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fraction:
\[ y(x^2 + x + 4) = x^2 - x + 4 \]

### Step 2: Expand the Equation
Next, we expand both sides:
\[ yx^2 + yx + 4y = x^2 - x + 4 \]

### Step 3: Rearrange the Equation
Now, we rearrange the equation to bring all terms to one side:
\[ yx^2 - x^2 + yx + x + 4y - 4 = 0 \]
This simplifies to:
\[ (y - 1)x^2 + (y + 1)x + (4y - 4) = 0 \]

### Step 4: Identify Coefficients
In this quadratic equation in \( x \), we identify the coefficients:
- \( a = y - 1 \)
- \( b = y + 1 \)
- \( c = 4y - 4 \)

### Step 5: Apply the Discriminant Condition
For \( x \) to be real, the discriminant \( D \) of the quadratic must be non-negative:
\[ D = b^2 - 4ac \geq 0 \]
Substituting the coefficients:
\[ (y + 1)^2 - 4(y - 1)(4y - 4) \geq 0 \]

### Step 6: Expand the Discriminant
Expanding the discriminant:
\[ (y + 1)^2 - 4[(y - 1)(4y - 4)] \geq 0 \]
Calculating \( (y + 1)^2 \):
\[ y^2 + 2y + 1 \]
Calculating \( 4(y - 1)(4y - 4) \):
\[ 4(y - 1)(4y - 4) = 16y^2 - 16y - 16 \]
So, the discriminant becomes:
\[ y^2 + 2y + 1 - (16y^2 - 16y - 16) \geq 0 \]
This simplifies to:
\[ y^2 + 2y + 1 - 16y^2 + 16y + 16 \geq 0 \]
Combining like terms:
\[ -15y^2 + 18y + 17 \geq 0 \]

### Step 7: Multiply by -1
Multiplying the entire inequality by -1 (which reverses the inequality):
\[ 15y^2 - 18y - 17 \leq 0 \]

### Step 8: Factor the Quadratic
Now we need to factor \( 15y^2 - 18y - 17 \). We can use the quadratic formula to find the roots:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 15 \cdot (-17)}}{2 \cdot 15} \]
Calculating the discriminant:
\[ 324 + 1020 = 1344 \]
Finding the roots:
\[ y = \frac{18 \pm \sqrt{1344}}{30} \]
Calculating \( \sqrt{1344} \):
\[ \sqrt{1344} = 36.66 \text{ (approximately)} \]
Thus, the roots are:
\[ y_1 \approx \frac{54.66}{30} \approx 1.822 \quad \text{and} \quad y_2 \approx \frac{-18.66}{30} \approx -0.622 \]

### Step 9: Determine the Range
The quadratic \( 15y^2 - 18y - 17 \) opens upwards (since the coefficient of \( y^2 \) is positive). The values of \( y \) for which this expression is less than or equal to zero are between the roots:
\[ y \in \left[-0.622, 1.822\right] \]

### Final Answer
Thus, the range of the rational expression is:
\[ \boxed{\left[-\frac{3}{5}, \frac{5}{3}\right]} \]