The number of integral values of `m` for which the equation `(1+m^(2)) x^(2) - 2(1+3m)x+(1+8m) = 0`, has no real roots is

Options

A. `3`
B. `1`
C. `2`
D. infinitely many

Explanation

To determine the number of integral values of \( m \) for which the equation

\[ (1+m^2)x^2 - 2(1+3m)x + (1+8m) = 0 \]

has no real roots, we need to analyze the discriminant of the quadratic equation. The discriminant \( D \) must be less than 0 for the quadratic to have no real roots.

### Step 1: Identify coefficients
The coefficients of the quadratic equation are:
- \( a = 1 + m^2 \)
- \( b = -2(1 + 3m) \)
- \( c = 1 + 8m \)

### Step 2: Write the discriminant
The discriminant \( D \) is given by the formula:

\[ D = b^2 - 4ac \]

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = [-2(1 + 3m)]^2 - 4(1 + m^2)(1 + 8m) \]

### Step 3: Simplify the discriminant
Calculating \( b^2 \):

\[ b^2 = 4(1 + 3m)^2 = 4(1 + 6m + 9m^2) = 4 + 24m + 36m^2 \]

Now calculating \( 4ac \):

\[ 4ac = 4(1 + m^2)(1 + 8m) = 4[(1 + 8m) + m^2(1 + 8m)] = 4(1 + 8m + m^2 + 8m^3) \]

This expands to:

\[ 4 + 32m + 4m^2 + 32m^3 \]

### Step 4: Set up the inequality
Now substituting back into the discriminant:

\[ D = (4 + 24m + 36m^2) - (4 + 32m + 4m^2 + 32m^3) \]

This simplifies to:

\[ D = 36m^2 - 4m^2 + 24m - 32m - 32m^3 = 32m^2 - 32m^3 - 8m \]

### Step 5: Factor the expression
Factoring out \( -8m \):

\[ D = -8m(4m^2 + 4m + 1) \]

### Step 6: Analyze the inequality
For the quadratic \( 4m^2 + 4m + 1 \):

This is always positive since its discriminant \( 4 - 16 = -12 < 0 \). Therefore, \( D < 0 \) when \( -8m < 0 \), which implies:

\[ m > 0 \]

### Step 7: Count integral values
Since \( m \) must be an integer greater than 0, the integral values of \( m \) are \( 1, 2, 3, \ldots \) which means there are infinitely many integral values of \( m \).

### Conclusion
The number of integral values of \( m \) for which the equation has no real roots is:

\[ \text{Infinitely many} \]