`ax^2 + bx + c = 0(a > 0),` has two roots `alpha and beta` such `alpha < -2 and beta > 2,` then

Options

A. ` a - |b| + c lt 0 `
B. ` c lt 0, b^(2) - 4ac gt 0 `
C. `4a - 2 |b| + c lt 0 `
D. ` 9a - 3 |b| + c lt 0 `

Explanation

To solve the equation \( ax^2 + bx + c = 0 \) with the conditions given, we will analyze the roots \( \alpha \) and \( \beta \) based on the properties of quadratic equations.

### Step 1: Understanding the Quadratic Equation
The quadratic equation is given by:
\[ f(x) = ax^2 + bx + c \]
where \( a > 0 \). This means the parabola opens upwards.

### Step 2: Analyzing the Roots
We know that:
- \( \alpha < -2 \)
- \( \beta > 2 \)

This indicates that the graph of the quadratic function must cross the x-axis at two points, one to the left of -2 and one to the right of 2.

### Step 3: Determining the Nature of Roots
For the quadratic equation to have two distinct real roots, the discriminant must be greater than zero:
\[ D = b^2 - 4ac > 0 \]

### Step 4: Evaluating the Value of \( c \)
Since the parabola opens upwards and has roots \( \alpha \) and \( \beta \), the vertex of the parabola must be located between the two roots. The vertex \( x_v \) is given by:
\[ x_v = -\frac{b}{2a} \]
Since \( \alpha < -2 \) and \( \beta > 2 \), it follows that \( -2 < x_v < 2 \).

### Step 5: Finding \( f(0) \)
The value of \( f(0) \) gives us the y-intercept, which is \( c \):
\[ f(0) = c \]
Since the graph is below the x-axis between the roots, we know that \( c < 0 \).

### Step 6: Analyzing the Conditions for \( a, b, c \)
We need to check the conditions for \( a + b + c < 0 \) and \( a - 2b + c < 0 \):
1. **Condition 1**: \( a + b + c < 0 \)
2. **Condition 2**: \( a - 2b + c < 0 \)

### Step 7: Conclusion
From the analysis, we can conclude that:
- The discriminant \( b^2 - 4ac > 0 \) ensures two distinct roots.
- The y-intercept \( c < 0 \) ensures the graph is below the x-axis at \( x = 0 \).
- The conditions involving \( a, b, c \) must be checked to determine which options are valid.

### Final Answer
Based on the analysis, the correct options can be determined from the conditions derived above.