The number of integral value of `a,a, in [-5, 5]` for which the equation: `x ^(2) +2 (a-1) x+a+5=0` has one root smalleer than 1 and the other root greater than 3 is :

Answer

To solve the problem, we need to analyze the quadratic equation given by:

\[ x^2 + 2(a-1)x + (a+5) = 0 \]

We want to find the integral values of \( a \) in the range \([-5, 5]\) such that one root is smaller than 1 and the other root is greater than 3.

### Step 1: Identify the coefficients
The coefficients of the quadratic equation are:
- \( A = 1 \)
- \( B = 2(a-1) = 2a - 2 \)
- \( C = a + 5 \)

### Step 2: Conditions for the roots
Let the roots of the quadratic equation be \( r_1 \) and \( r_2 \). We need:
1. \( r_1 < 1 \)
2. \( r_2 > 3 \)

### Step 3: Use Vieta's formulas
From Vieta's formulas, we know:
- The sum of the roots \( r_1 + r_2 = -\frac{B}{A} = -\frac{2a - 2}{1} = 2 - 2a \)
- The product of the roots \( r_1 r_2 = \frac{C}{A} = a + 5 \)

### Step 4: Set up the inequalities
To satisfy the conditions for the roots:
1. For \( r_1 < 1 \):
\[ r_1 + r_2 < 1 + r_2 \Rightarrow 2 - 2a < 1 + r_2 \Rightarrow r_2 > 1 - 2 + 2a \Rightarrow r_2 > 2a - 1 \]

2. For \( r_2 > 3 \):
\[ r_1 + r_2 > r_1 + 3 \Rightarrow 2 - 2a > r_1 + 3 \Rightarrow r_1 < 2 - 2a - 3 \Rightarrow r_1 < -1 - 2a \]

### Step 5: Analyze the discriminant
The quadratic must have real roots, so the discriminant \( D \) must be non-negative:
\[ D = B^2 - 4AC = (2a - 2)^2 - 4(1)(a + 5) \geq 0 \]
Expanding this:
\[ (2a - 2)^2 - 4(a + 5) = 4a^2 - 8a + 4 - 4a - 20 \geq 0 \]
\[ 4a^2 - 12a - 16 \geq 0 \]
Dividing by 4:
\[ a^2 - 3a - 4 \geq 0 \]
Factoring:
\[ (a - 4)(a + 1) \geq 0 \]
The roots are \( a = 4 \) and \( a = -1 \). The intervals where the inequality holds are:
- \( a \leq -1 \) or \( a \geq 4 \)

### Step 6: Combine conditions
We also need \( a \) to be in the range \([-5, 5]\):
- From \( a \leq -1 \): valid values are \( -5, -4, -3, -2 \) (4 values)
- From \( a \geq 4 \): valid value is \( 4 \) (1 value)

### Conclusion
The integral values of \( a \) that satisfy all conditions are:
- \( -5, -4, -3, -2, 4 \)

Thus, the total number of integral values of \( a \) is **5**.