Mathematics

The key insight is that this product equals zero when either factor equals zero, so you need to count how many distinct real roots come from both quadratics combined. For the equation to have exactly two distinct real roots total, you must carefully track when each quadratic contributes real roots and ensure they don't overlap. Use the discriminant to determine when each factor has real solutions, then find the range of $m$ where the total count is exactly two—this often means one quadratic has two real roots while the other has none, or both share exactly one common root.

The key insight is that this product equals zero when either factor equals zero, so you need to count how many distinct real roots come from both quadratics combined. For the equation to have exactly two distinct real roots total, you must carefully track when each quadratic contributes real roots and ensure they don't overlap. Use the discriminant to determine when each factor has real solutions, then find the range of $m$ where the total count is exactly two—this often means one quadratic has two real roots while the other has none, or both share exactly one common root.