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
such that one root is greater than 3, and the other is smaller than 3.

Answer

To solve the problem of finding the values of the parameter \( a \) for which the roots of the quadratic equation \( x^2 + 2(a-1)x + (a+5) = 0 \) have one root greater than 3 and the other root smaller than 3, we will follow these steps:

### Step 1: Identify the Quadratic Equation
The given quadratic equation is:
\[ f(x) = x^2 + 2(a-1)x + (a+5) \]

### Step 2: Determine the Conditions for the Roots
For one root to be greater than 3 and the other to be less than 3, we need:
1. \( f(3) < 0 \) (the parabola must be below the x-axis at \( x = 3 \)).
2. The discriminant \( D \) of the quadratic must be greater than 0 (for the roots to be real and distinct).

### Step 3: Calculate \( f(3) \)
Substituting \( x = 3 \) into the quadratic equation:
\[ f(3) = 3^2 + 2(a-1)(3) + (a+5) \]
\[ = 9 + 6(a-1) + (a+5) \]
\[ = 9 + 6a - 6 + a + 5 \]
\[ = 7a + 8 \]
We require \( f(3) < 0 \):
\[ 7a + 8 < 0 \]
\[ 7a < -8 \]
\[ a < -\frac{8}{7} \]

### Step 4: Calculate the Discriminant
The discriminant \( D \) of the quadratic equation is given by:
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = 2(a-1) \), and \( c = a + 5 \):
\[ D = [2(a-1)]^2 - 4(1)(a+5) \]
\[ = 4(a-1)^2 - 4(a+5) \]
\[ = 4[(a-1)^2 - (a+5)] \]
Expanding:
\[ = 4[a^2 - 2a + 1 - a - 5] \]
\[ = 4[a^2 - 3a - 4] \]
We require \( D > 0 \):
\[ a^2 - 3a - 4 > 0 \]

### Step 5: Factor the Quadratic
To factor \( a^2 - 3a - 4 \):
\[ = (a - 4)(a + 1) > 0 \]
The roots are \( a = 4 \) and \( a = -1 \).

### Step 6: Analyze the Sign of the Factors
Using a number line, we analyze the intervals:
1. \( (-\infty, -1) \)
2. \( (-1, 4) \)
3. \( (4, \infty) \)

Testing intervals:
- For \( a < -1 \): both factors are negative, product is positive.
- For \( -1 < a < 4 \): one factor is negative, the other is positive, product is negative.
- For \( a > 4 \): both factors are positive, product is positive.

Thus, \( a^2 - 3a - 4 > 0 \) for:
\[ a < -1 \quad \text{or} \quad a > 4 \]

### Step 7: Combine Conditions
We have two conditions:
1. \( a < -\frac{8}{7} \)
2. \( a < -1 \) or \( a > 4 \)

The intersection of these conditions gives:
- From \( a < -1 \): valid as \( -1 < -\frac{8}{7} \).
- The valid range is \( (-\infty, -\frac{8}{7}) \).

### Final Answer
Thus, the values of \( a \) for which one root is greater than 3 and the other is smaller than 3 are:
\[ \boxed{(-\infty, -\frac{8}{7})} \]