1. **State the problem:** Solve the equation $ (3x - 8)(4x - 5) = 0 $ for $x$. 2. **Formula and rule:** If a product of two factors equals zero, then at least one of the factors must be zero. This is called the zero product property: $$ A \times B = 0 \implies A = 0 \text{ or } B = 0 $$ 3. **Apply the zero product property:** Set each factor equal to zero: $$ 3x - 8 = 0 \quad \text{or} \quad 4x - 5 = 0 $$ 4. **Solve each equation:** For $3x - 8 = 0$: $$ 3x = 8 $$ $$ x = \frac{8}{3} $$ For $4x - 5 = 0$: $$ 4x = 5 $$ $$ x = \frac{5}{4} $$ 5. **Final answer:** $$ x = \frac{8}{3} \quad \text{or} \quad x = \frac{5}{4} $$ These are the two solutions to the equation.