1. **State the problem:** Find the two solutions to the equation $$(5x - 1)(x - 2) = 0.$$\n\n2. **Recall the zero product property:** If a product of two factors equals zero, then at least one of the factors must be zero. That is, if $$A \cdot B = 0,$$ then either $$A = 0$$ or $$B = 0.$$\n\n3. **Apply the zero product property:** Set each factor equal to zero:\n$$5x - 1 = 0$$\n$$x - 2 = 0.$$\n\n4. **Solve each equation:**\nFor $$5x - 1 = 0$$:\nAdd 1 to both sides:\n$$5x - \cancel{1} + 1 = \cancel{0} + 1 \Rightarrow 5x = 1.$$\nDivide both sides by 5:\n$$\frac{\cancel{5}x}{\cancel{5}} = \frac{1}{5} \Rightarrow x = \frac{1}{5}.$$\n\nFor $$x - 2 = 0$$:\nAdd 2 to both sides:\n$$x - \cancel{2} + 2 = \cancel{0} + 2 \Rightarrow x = 2.$$\n\n5. **Final answer:** The two solutions are $$x = \frac{1}{5}$$ and $$x = 2.$$