Every Factor Gets a Turn

The rule scales. x(x - 3)(x + 4) = 0 holds three factors, so the split makes three cases: x = 0, x - 3 = 0, x + 4 = 0. Three solutions: 0, 3 and -4.

Nothing new is happening: a product of three things is zero exactly when one of them is. The rule you learned for two factors was never about the number two.

Equations like this come from factoring cubics, the chapter six shapes: pull the x, split the trinomial, and a three-factor product stands ready for a three-way split. The count of solutions matches the count of factors, each factor taking its turn.