The number of positive integral values of `m`, `m le 16` for which the equation `(x^(2) +x+1) ^(2) - (m-3)(x^(2) +x+1) +m=0,` has 4 distinct real root is:

Answer

To solve the problem, we need to find the number of positive integral values of \( m \) (where \( m \leq 16 \)) for which the equation

\[ (x^2 + x + 1)^2 - (m-3)(x^2 + x + 1) + m = 0 \]

has 4 distinct real roots.

### Step 1: Substitute \( t = x^2 + x + 1 \)

Let \( t = x^2 + x + 1 \). Then the equation becomes:

\[ t^2 - (m-3)t + m = 0 \]

### Step 2: Analyze the quadratic in \( t \)

For the quadratic equation \( t^2 - (m-3)t + m = 0 \) to have real roots, the discriminant must be greater than or equal to zero:

\[ D = (m-3)^2 - 4 \cdot 1 \cdot m \geq 0 \]

### Step 3: Calculate the discriminant

Calculating the discriminant:

\[ D = (m-3)^2 - 4m = m^2 - 6m + 9 - 4m = m^2 - 10m + 9 \]

Setting the discriminant greater than or equal to zero:

\[ m^2 - 10m + 9 \geq 0 \]

### Step 4: Solve the quadratic inequality

To find the roots of the quadratic equation \( m^2 - 10m + 9 = 0 \):

Using the quadratic formula:

\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1} = \frac{10 \pm \sqrt{100 - 36}}{2} = \frac{10 \pm \sqrt{64}}{2} = \frac{10 \pm 8}{2} \]

Thus, the roots are:

\[ m = \frac{18}{2} = 9 \quad \text{and} \quad m = \frac{2}{2} = 1 \]

### Step 5: Analyze the intervals

The quadratic \( m^2 - 10m + 9 \) opens upwards (since the coefficient of \( m^2 \) is positive). Therefore, the inequality \( m^2 - 10m + 9 \geq 0 \) holds true for:

\[ m \leq 1 \quad \text{or} \quad m \geq 9 \]

### Step 6: Find the range of \( m \)

Since we are interested in positive integral values of \( m \) such that \( m \leq 16 \), we consider:

1. \( m = 1 \) (from \( m \leq 1 \))
2. \( m = 9, 10, 11, 12, 13, 14, 15, 16 \) (from \( m \geq 9 \))

### Step 7: Count the valid values of \( m \)

The valid positive integral values of \( m \) are:

- From \( m \leq 1 \): \( 1 \)
- From \( m \geq 9 \): \( 9, 10, 11, 12, 13, 14, 15, 16 \)

Thus, the total number of valid values of \( m \) is:

\[ 1 + 8 = 9 \]

### Conclusion

The number of positive integral values of \( m \) such that the equation has 4 distinct real roots is **9**.