The number of negative intergral values of m for which the expression `x ^(2) + 2(m -1) x +m +5` is positive `AA x gt 1` is:

Answer

To solve the problem, we need to determine the number of negative integral values of \( m \) for which the expression \( x^2 + 2(m - 1)x + (m + 5) \) is positive for all \( x > 1 \).

### Step-by-Step Solution:

1. **Identify the Quadratic Expression**:
The given expression is:
\[ f(x) = x^2 + 2(m - 1)x + (m + 5) \]
This is a quadratic function in \( x \).

2. **Condition for the Quadratic to be Positive**:
For the quadratic to be positive for all \( x > 1 \), the coefficient of \( x^2 \) must be positive (which it is, as it equals 1), and the discriminant must be negative.

3. **Calculate the Discriminant**:
The discriminant \( D \) of a quadratic \( ax^2 + bx + c \) is given by:
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = 2(m - 1) \), and \( c = m + 5 \). Therefore, we have:
\[ D = [2(m - 1)]^2 - 4 \cdot 1 \cdot (m + 5) \]
Simplifying this:
\[ D = 4(m - 1)^2 - 4(m + 5) \]
\[ D = 4[(m - 1)^2 - (m + 5)] \]

4. **Set the Discriminant Less Than Zero**:
We need:
\[ (m - 1)^2 - (m + 5) < 0 \]
Expanding this:
\[ (m^2 - 2m + 1) - m - 5 < 0 \]
\[ m^2 - 3m - 4 < 0 \]

5. **Factor the Quadratic**:
We can factor \( m^2 - 3m - 4 \):
\[ (m + 1)(m - 4) < 0 \]

6. **Determine the Intervals**:
The roots of the equation are \( m = -1 \) and \( m = 4 \). The quadratic will be negative between the roots:
\[ -1 < m < 4 \]

7. **Identify the Integral Values of \( m \)**:
The integral values of \( m \) in the interval \( (-1, 4) \) are:
\[ 0, 1, 2, 3 \]
These are all the integral values of \( m \) that satisfy the condition.

8. **Count the Negative Integral Values**:
The negative integral values of \( m \) in the interval \( (-1, 4) \) are none. Thus, the number of negative integral values of \( m \) is:
\[ \text{Number of negative integral values of } m = 0 \]

### Final Answer:
The number of negative integral values of \( m \) for which the expression is positive for all \( x > 1 \) is \( 0 \).