If the equation `ax^(2)+bx+c=0 (a gt 0)` has two roots `alpha and beta` such that `alpha lt -2 and beta gt 2`, then

Options

A. `b^(2) -4ac gt 0`
B. `4a+2 |b| +c lt 0`
C. `a+|b| +c lt 0`
D. `c lt 0`

Answer

To solve the problem, we need to analyze the quadratic equation given by \( ax^2 + bx + c = 0 \) where \( a > 0 \). The roots of this equation are denoted as \( \alpha \) and \( \beta \) with the conditions \( \alpha < -2 \) and \( \beta > 2 \).

### Step-by-Step Solution:

1. **Understanding the Roots**:
- The roots \( \alpha \) and \( \beta \) are the solutions to the quadratic equation. Given that \( \alpha < -2 \) and \( \beta > 2 \), we can infer that the graph of the quadratic function opens upwards (since \( a > 0 \)) and crosses the x-axis at two points, one to the left of -2 and one to the right of 2.

2. **Using Vieta's Formulas**:
- According to Vieta's formulas, for the quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

3. **Analyzing the Sum of Roots**:
- Since \( \alpha < -2 \) and \( \beta > 2 \), we can express the sum:
\[ \alpha + \beta < -2 + 2 = 0 \]
- Therefore, we have \( -\frac{b}{a} < 0 \), which implies that \( b > 0 \).

4. **Analyzing the Product of Roots**:
- The product of the roots can be analyzed as follows:
\[ \alpha \beta < (-2)(2) = -4 \]
- Thus, \( \frac{c}{a} < -4 \) which leads to \( c < -4a \) (since \( a > 0 \)).

5. **Conclusion**:
- From the analysis, we conclude that:
- \( b > 0 \)
- \( c < -4a \)

### Final Result:
The conditions on the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \) are:
- \( b > 0 \)
- \( c < -4a \)