Intersection of Line and Circle given Midpoint Coordinate

To find the correct range for $m$, we substitute the equation of the line $y = mx + 1$ into the circle $(x - 3)^2 + (y + 2)^2 = 25$ to form a quadratic equation in $x$. Using the relation for the sum of the roots, we set the $x$-coordinate of the midpoint $\frac{x_1 + x_2}{2}$ equal to $-\frac{3}{5}$, which yields the quadratic equation $m^2 - 5m + 6 = 0$. Solving this gives $m = 2$ and $m = 3$, both of which ensure two distinct points of intersection as the discriminant is strictly positive. Consequently, $m$ lies in the interval $[2, 4)$, making option (B) the correct choice.