To find the line of reflection, we need to determine the line that maps the function \( f(x) = x^2 + 5x - 6 \) to its reflection \( f'(x) = -x^2 - 5x + 6 \).
We can find the line of reflection by considering that reflecting \( f(x) \) over a certain line results in \( f'(x) \). The line of reflection, denoted as \( y = mx + b \), typically bisects the vertical distance between \( f(x) \) and \( f'(x) \) at every point.
Let's first find the average of \( f(x) \) and \( f'(x) \):
\[ f(x) + f'(x) = (x^2 + 5x - 6) + (-x^2 - 5x + 6) = (x^2 - x^2) + (5x - 5x) + (-6 + 6) = 0 \]
The two functions sum up to zero at every point. Therefore, the mean will also be zero at every point, which suggests that they are symmetric around the horizontal line \( y = 0 \).
That means the line of reflection is:
\[ y = 0 \]
Thus, the equation for the line of reflection is:
\[ \boxed{y = 0} \]