1. **State the problem:** Expand and solve the equation $5x(x-1) = 0$. 2. **Expand the expression:** Use the distributive property $a(b+c) = ab + ac$. $$5x(x-1) = 5x \cdot x - 5x \cdot 1 = 5x^2 - 5x$$ 3. **Set the equation to zero:** $$5x^2 - 5x = 0$$ 4. **Factor the equation:** Factor out the common term $5x$. $$5x(x - 1) = 0$$ 5. **Solve for $x$:** Set each factor equal to zero. - $5x = 0 \Rightarrow x = 0$ - $x - 1 = 0 \Rightarrow x = 1$ 6. **Final answer:** The solutions to the equation are $x = 0$ and $x = 1$.