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 determine the number of integral values of \( a \) in the interval \([-5, 5]\) for which the quadratic equation

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

has one root smaller than 1 and the other root greater than 3.

### Step 1: Identify the coefficients

The quadratic equation can be rewritten in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = 2(a - 1) \)
- \( c = a + 5 \)

### Step 2: Conditions for the roots

For the quadratic to have one root smaller than 1 and the other root greater than 3, we need to satisfy the following conditions:
1. The value of the quadratic at \( x = 1 \) must be negative: \( f(1) < 0 \).
2. The value of the quadratic at \( x = 3 \) must also be negative: \( f(3) < 0 \).
3. The discriminant must be positive for the roots to be real and distinct: \( D > 0 \).

### Step 3: Calculate \( f(1) < 0 \)

Calculating \( f(1) \):

\[ f(1) = 1^2 + 2(a - 1)(1) + (a + 5) = 1 + 2(a - 1) + (a + 5) \]
\[ = 1 + 2a - 2 + a + 5 = 3a + 4 \]
Setting the inequality:

\[ 3a + 4 < 0 \implies 3a < -4 \implies a < -\frac{4}{3} \]

### Step 4: Calculate \( f(3) < 0 \)

Calculating \( f(3) \):

\[ f(3) = 3^2 + 2(a - 1)(3) + (a + 5) = 9 + 6(a - 1) + (a + 5) \]
\[ = 9 + 6a - 6 + a + 5 = 7a + 8 \]
Setting the inequality:

\[ 7a + 8 < 0 \implies 7a < -8 \implies a < -\frac{8}{7} \]

### Step 5: Calculate the discriminant \( D > 0 \)

The discriminant \( D \) is given by:

\[ D = b^2 - 4ac = (2(a - 1))^2 - 4(1)(a + 5) \]
\[ = 4(a - 1)^2 - 4(a + 5) = 4(a^2 - 2a + 1 - a - 5) = 4(a^2 - 3a - 4) \]
Setting the inequality:

\[ 4(a^2 - 3a - 4) > 0 \implies a^2 - 3a - 4 > 0 \]

Factoring the quadratic:

\[ (a - 4)(a + 1) > 0 \]

### Step 6: Solve the inequality \( (a - 4)(a + 1) > 0 \)

Using a number line, we find the intervals where the product is positive:

- The critical points are \( a = -1 \) and \( a = 4 \).
- The intervals are \( (-\infty, -1) \) and \( (4, \infty) \).

### Step 7: Combine the conditions

We have three conditions:
1. \( a < -\frac{4}{3} \)
2. \( a < -\frac{8}{7} \)
3. \( a < -1 \) or \( a > 4 \)

The most restrictive condition is \( a < -\frac{8}{7} \). Therefore, we consider the intersection of the intervals:

- From \( a < -\frac{8}{7} \) and \( (-\infty, -1) \), we take \( (-\infty, -\frac{8}{7}) \).

### Step 8: Determine integral values in the interval \([-5, 5]\)

The integral values of \( a \) in the interval \([-5, 5]\) that satisfy \( a < -\frac{8}{7} \) are:

- \( -5, -4, -3, -2 \)

### Conclusion

Thus, the number of integral values of \( a \) in the interval \([-5, 5]\) for which the quadratic equation has one root smaller than 1 and the other root greater than 3 is:

\[ \boxed{4} \]