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 need to analyze the discriminant of the quadratic equation. The roots of a quadratic equation are not real if the discriminant is less than zero.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = 2(a-1) \)
- \( c = a + 5 \)

2. **Write the discriminant**:
The discriminant \( D \) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = [2(a-1)]^2 - 4(1)(a + 5) \]

3. **Simplify the discriminant**:
Expanding \( D \):
\[ D = 4(a-1)^2 - 4(a + 5) \]
\[ D = 4(a^2 - 2a + 1) - 4a - 20 \]
\[ D = 4a^2 - 8a + 4 - 4a - 20 \]
\[ D = 4a^2 - 12a - 16 \]

4. **Set the discriminant less than zero**:
To find when the roots are not real, we set the discriminant \( D < 0 \):
\[ 4a^2 - 12a - 16 < 0 \]

5. **Divide by 4**:
Simplifying the inequality:
\[ a^2 - 3a - 4 < 0 \]

6. **Factor the quadratic**:
We can factor the quadratic expression:
\[ (a - 4)(a + 1) < 0 \]

7. **Determine the intervals**:
To solve the inequality \( (a - 4)(a + 1) < 0 \), we find the critical points where the expression equals zero:
- \( a - 4 = 0 \) gives \( a = 4 \)
- \( a + 1 = 0 \) gives \( a = -1 \)

The critical points divide the number line into intervals:
- \( (-\infty, -1) \)
- \( (-1, 4) \)
- \( (4, \infty) \)

8. **Test the intervals**:
We can test a point from each interval to see where the product is negative:
- For \( a = -2 \) (in \( (-\infty, -1) \)): \( (-2 - 4)(-2 + 1) = (-6)(-1) > 0 \)
- For \( a = 0 \) (in \( (-1, 4) \)): \( (0 - 4)(0 + 1) = (-4)(1) < 0 \)
- For \( a = 5 \) (in \( (4, \infty) \)): \( (5 - 4)(5 + 1) = (1)(6) > 0 \)

The inequality \( (a - 4)(a + 1) < 0 \) holds true in the interval \( (-1, 4) \).

### Conclusion:
The values of the parameter \( a \) for which the roots of the quadratic equation are not real are:
\[ \boxed{(-1, 4)} \]