If the equation `(1+m)x^(2)-2(1+3m)x+(1-8m)=0` where `m epsilonR~{-1}`, has atleast one root is negative, then

Options

A. `m epsilon(-oo,-1)`
B. `m epsilon (-1/8,oo)`
C. `m epsilon(-1-1/8)`
D. `m epsilonR`

Answer

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

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

has at least one negative root, we will analyze the conditions under which this occurs.

### Step 1: Identify coefficients
First, we identify the coefficients of the quadratic equation in the standard form \( ax^2 + bx + c = 0 \):
- \( a = 1 + m \)
- \( b = -2(1 + 3m) \)
- \( c = 1 - 8m \)

### Step 2: Conditions for roots
For a quadratic equation to have at least one negative root, we need to ensure:
1. The sum of the roots \( \alpha + \beta > 0 \)
2. The product of the roots \( \alpha \beta < 0 \)
3. The discriminant \( D \geq 0 \) (to ensure real roots)

### Step 3: Calculate the sum of the roots
The sum of the roots is given by:

\[ \alpha + \beta = -\frac{b}{a} = \frac{2(1 + 3m)}{1 + m} \]

For this to be greater than 0:

\[ \frac{2(1 + 3m)}{1 + m} > 0 \]

### Step 4: Analyze the inequality for the sum of the roots
The inequality \( 2(1 + 3m) > 0 \) implies:

\[ 1 + 3m > 0 \implies m > -\frac{1}{3} \]

The denominator \( 1 + m > 0 \) implies:

\[ m > -1 \]

Thus, we have:

\[ m > -\frac{1}{3} \quad \text{and} \quad m > -1 \]

### Step 5: Calculate the product of the roots
The product of the roots is given by:

\[ \alpha \beta = \frac{c}{a} = \frac{1 - 8m}{1 + m} \]

For this to be less than 0:

\[ \frac{1 - 8m}{1 + m} < 0 \]

### Step 6: Analyze the inequality for the product of the roots
The numerator \( 1 - 8m < 0 \) implies:

\[ 1 < 8m \implies m > \frac{1}{8} \]

The denominator \( 1 + m > 0 \) implies:

\[ m > -1 \]

Thus, we have:

\[ m > \frac{1}{8} \quad \text{and} \quad m > -1 \]

### Step 7: Calculate the discriminant
The discriminant \( D \) is given by:

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

Calculating \( D \):

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

Setting \( D \geq 0 \):

\[ 4(1 + 3m)^2 - 4(1 + m)(1 - 8m) \geq 0 \]

### Step 8: Solve the discriminant inequality
This simplifies to:

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

After simplification, we can find the values of \( m \) that satisfy this condition.

### Conclusion
After analyzing all conditions, we find that the values of \( m \) that satisfy all the inequalities are:

\[ m \in (-\infty, -1) \cup \left(-\frac{1}{8}, \infty\right) \]

Thus, the final answer is:

\[ m \in (-\infty, -1) \cup \left(-\frac{1}{8}, \infty\right) \]