Dividing by variable when solving algebraic equation

[FallentoReason]

To be more exact, the reason the answer changes is because you're actually altering the original equation. The equation x(x+1) = 0 is a quadratic equation i.e. x^2 + x = 0. This is the form of a quadratic equation, which usually looks like ax^2 + bx + c = 0. Quadratics will always have a pair of solutions (and you can clearly see this when they are graphed - they intersect the x axis twice). But upon dividing by x, you are now left with a linear equation, and linear equations will only intersect the x axis once, hence only one solution.

EDIT: sorry, it's not entirely true that quadratics will always have two solutions. They can have one, and this happens when the curve is tangent to the x axis, meaning it's touching it exactly once.