Find the values of the parameter a for which the roots of the quadratic equation `x^(2)+2(a-1)x+a+5=0` are
not real

Answer

To find the values of the parameter \( a \) for which the roots of the quadratic equation \( x^2 + 2(a-1)x + (a+5) = 0 \) are not real, we will analyze the discriminant of the quadratic equation.

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \( Ax^2 + Bx + C = 0 \). Here, we have:
- \( A = 1 \)
- \( B = 2(a-1) \)
- \( C = a + 5 \)

### Step 2: Write the discriminant
The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
\[ D = B^2 - 4AC \]
For the roots to be not real, the discriminant must be less than 0:
\[ D < 0 \]

### Step 3: Substitute the coefficients into the discriminant formula
Substituting \( A \), \( B \), and \( C \) into the discriminant formula, we get:
\[ D = [2(a-1)]^2 - 4(1)(a+5) \]

### Step 4: Simplify the discriminant
Calculating \( D \):
\[ D = 4(a-1)^2 - 4(a+5) \]
\[ D = 4[(a-1)^2 - (a+5)] \]

### Step 5: Expand and simplify further
Expanding \( (a-1)^2 \):
\[ (a-1)^2 = a^2 - 2a + 1 \]
Thus,
\[ D = 4[a^2 - 2a + 1 - a - 5] \]
\[ D = 4[a^2 - 3a - 4] \]

### Step 6: Set the discriminant less than zero
To find the values of \( a \) for which the roots are not real, we set the discriminant less than 0:
\[ 4(a^2 - 3a - 4) < 0 \]
Dividing both sides by 4 (since 4 is positive):
\[ a^2 - 3a - 4 < 0 \]

### Step 7: Factor the quadratic expression
Now we factor the quadratic expression:
\[ a^2 - 3a - 4 = (a - 4)(a + 1) \]
Thus, we need to solve:
\[ (a - 4)(a + 1) < 0 \]

### Step 8: Determine the intervals
The roots of the equation \( (a - 4)(a + 1) = 0 \) are \( a = 4 \) and \( a = -1 \). We will analyze the sign of the product in the intervals:
1. \( a < -1 \)
2. \( -1 < a < 4 \)
3. \( a > 4 \)

### Step 9: Test the intervals
- For \( a < -1 \) (e.g., \( a = -2 \)): \( (-2 - 4)(-2 + 1) = (-6)(-1) > 0 \)
- For \( -1 < a < 4 \) (e.g., \( a = 0 \)): \( (0 - 4)(0 + 1) = (-4)(1) < 0 \)
- For \( a > 4 \) (e.g., \( a = 5 \)): \( (5 - 4)(5 + 1) = (1)(6) > 0 \)

### Step 10: Conclusion
The product \( (a - 4)(a + 1) < 0 \) holds true for:
\[ -1 < a < 4 \]
Thus, the values of the parameter \( a \) for which the roots of the quadratic equation are not real are:
\[ \boxed{(-1, 4)} \]