If the equation `ax^2+bx+c=0(a>0)` has two roots α and β such that `α<−2` and `β>2` , then: a) `b^2-4ac>0` b) c<0 c) a+|b|+c>0 d) 4a+2|b|+c <0

Explanation

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

### Step-by-step Solution:

1. **Understanding the Roots**:
We know that \( \alpha < -2 \) and \( \beta > 2 \). This means that the roots of the quadratic equation are located on opposite sides of the y-axis.

2. **Using the Quadratic Formula**:
The roots of the quadratic equation can be expressed using the quadratic formula:
\[ \alpha, \beta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

3. **Condition for Real Roots**:
For the roots to be real and distinct, the discriminant must be positive:
\[ b^2 - 4ac > 0 \]
This confirms that option (a) is correct.

4. **Finding the Sign of \( c \)**:
We evaluate the quadratic at \( x = 0 \):
\[ f(0) = c \]
Since \( \alpha < -2 \) and \( \beta > 2 \), the function must cross the x-axis at some point between these roots. Therefore, \( f(0) < 0 \) implies:
\[ c < 0 \]
This confirms that option (b) is correct.

5. **Evaluating at \( x = 2 \)**:
We substitute \( x = 2 \) into the quadratic:
\[ f(2) = 4a + 2b + c \]
Since \( \beta > 2 \), the function must be negative at \( x = 2 \):
\[ 4a + 2b + c < 0 \]
We can rewrite this as:
\[ 4a + 2|b| + c < 0 \]
This confirms that option (d) is correct.

6. **Evaluating at \( x = -2 \)**:
Now we substitute \( x = -2 \):
\[ f(-2) = 4a - 2b + c \]
Since \( \alpha < -2 \), the function must be positive at \( x = -2 \):
\[ 4a - 2b + c > 0 \]
This does not confirm option (c) as correct.

### Conclusion:
The correct options based on the analysis are:
- (a) \( b^2 - 4ac > 0 \)
- (b) \( c < 0 \)
- (d) \( 4a + 2|b| + c < 0 \)