Step-by-step Solution:

The roots of the quadratic equation are \( m \) and \( n \). We are given the relationships: \[ m + n = \frac{2b}{a}, \quad mn = \frac{c}{a} \] Now, we need to evaluate: \[ \frac{b}{a(n^2 + c)} + \frac{b}{a(m^2 + c)} = \frac{b}{a} \left( \frac{1}{n^2 + \frac{c}{a}} + \frac{1}{m^2 + \frac{c}{a}} \right) \] This simplifies as follows: \[ = \frac{b}{a} \left( \frac{1}{n^2 + \frac{c}{a}} + \frac{1}{m^2 + \frac{c}{a}} \right) \] Next, express the terms in a form involving \( m \) and \( n \): \[ = \frac{m + n}{2} \left( \frac{1}{n^2 + mn} + \frac{1}{m^2 + mn} \right) \] This simplifies further: \[ = \frac{m + n}{2} \left( \frac{m + n}{mn(m + n)} \right) \] Finally, we get: \[ = \frac{m + n}{2mn} = \frac{b}{c}. \] Thus, the desired expression simplifies to: \[ \frac{b}{c}. \]