To solve the problem step by step, we start with the given equation \( ax^2 + bx + c = 0 \) and the condition provided:
1. **Understanding the Limit Condition**:
We are given that:
\[
\lim_{x \to m} \frac{|ax^2 + bx + c|}{ax^2 + bx + c} = 1
\]
This implies that as \( x \) approaches \( m \), the expression \( ax^2 + bx + c \) does not change sign, meaning it is either positive or negative but not zero.
2. **Analyzing the Expression**:
The limit condition can be rewritten as:
\[
|ax^2 + bx + c| = ax^2 + bx + c \quad \text{for } x \to m
\]
This indicates that \( ax^2 + bx + c \) must be greater than 0 when \( x \) approaches \( m \).
3. **Finding the Roots**:
The roots of the equation \( ax^2 + bx + c = 0 \) are \( \alpha \) and \( \beta \) where \( \alpha < \beta \). The quadratic will change signs at these roots.
4. **Determining the Sign of the Quadratic**:
- If \( a > 0 \), the parabola opens upwards. Therefore, \( ax^2 + bx + c > 0 \) outside the interval \( (\alpha, \beta) \) and \( ax^2 + bx + c < 0 \) within the interval.
- If \( a < 0 \), the parabola opens downwards. Thus, \( ax^2 + bx + c < 0 \) outside the interval \( (\alpha, \beta) \) and \( ax^2 + bx + c > 0 \) within the interval.
5. **Applying the Limit Condition**:
Since the limit condition states that \( ax^2 + bx + c \) is positive as \( x \to m \), we analyze the implications:
- If \( a > 0 \), \( m \) must lie outside the interval \( (\alpha, \beta) \) (either \( m < \alpha \) or \( m > \beta \)).
- If \( a < 0 \), \( m \) must lie within the interval \( (\alpha, \beta) \).
6. **Conclusion**:
Since the limit condition leads us to conclude that \( ax^2 + bx + c \) is positive, and given that \( a \) must be greater than 0 for this to hold true, we conclude that:
- \( a > 0 \) and \( m \) does not lie in the interval \( (\alpha, \beta) \).
Thus, the final answer is that \( a > 0 \) and \( m \) does not lie in \( (\alpha, \beta) \).
