Let a, b, c be real, if `ax^(2)+bx+c=0` has two real roots `alpha, beta,` where `alt-1 and betagt1`, then the value of `1+(c)/(a)+|(b)/(a)|` is :

Options

A. a. less than -1
B. b. greater than zero
C. c. equal to zero
D. d. equal to `b^(2)-4ac`

Answer

To solve the problem, we need to analyze the quadratic equation \( ax^2 + bx + c = 0 \) given the conditions on its roots \( \alpha \) and \( \beta \).

### Step-by-Step Solution:

1. **Understanding the Roots**:
We know that the roots \( \alpha \) and \( \beta \) satisfy \( \alpha < -1 \) and \( \beta > 1 \). This implies that the quadratic function must cross the x-axis at two points: one to the left of -1 and one to the right of 1.

2. **Evaluating the Function at Critical Points**:
We evaluate the quadratic function at \( x = -1 \) and \( x = 1 \):
- For \( x = -1 \):
\[ f(-1) = a(-1)^2 + b(-1) + c = a - b + c \]
- For \( x = 1 \):
\[ f(1) = a(1)^2 + b(1) + c = a + b + c \]

3. **Applying the Conditions**:
Since \( \alpha < -1 \) implies \( f(-1) > 0 \) (the function must be above the x-axis at this point):
\[ a - b + c > 0 \quad \text{(1)} \]
Since \( \beta > 1 \) implies \( f(1) < 0 \) (the function must be below the x-axis at this point):
\[ a + b + c < 0 \quad \text{(2)} \]

4. **Rearranging the Inequalities**:
From inequality (1):
\[ c > -a + b \quad \text{(3)} \]
From inequality (2):
\[ c < -a - b \quad \text{(4)} \]

5. **Combining the Inequalities**:
From (3) and (4), we can combine these inequalities:
\[ -a + b < c < -a - b \]

6. **Finding \( \frac{c}{a} + \left| \frac{b}{a} \right| \)**:
We want to find the value of:
\[ 1 + \frac{c}{a} + \left| \frac{b}{a} \right| \]
Let's denote \( \frac{c}{a} = k \), then we can express our inequalities in terms of \( k \):
\[ -1 + \frac{b}{a} < k < -1 - \frac{b}{a} \]

7. **Analyzing the Expression**:
If we denote \( \frac{b}{a} = m \), then we can rewrite the inequalities:
\[ -1 + m < k < -1 - m \]
This leads us to conclude that:
\[ k + |m| < -1 \]

8. **Final Calculation**:
From the inequalities, we can deduce:
\[ 1 + k + |m| < 0 \]
Thus, we find that:
\[ 1 + \frac{c}{a} + \left| \frac{b}{a} \right| = -1 \]

### Conclusion:
Therefore, the value of \( 1 + \frac{c}{a} + \left| \frac{b}{a} \right| \) is:
\[ \boxed{-1} \]