1. **State the problem:** Solve for $x$ in the equation $$(5x + 6)(3x - 8) = 0.$$ 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: $$a \cdot b = 0 \implies a = 0 \text{ or } b = 0.$$ 3. **Apply the property:** Set each factor equal to zero: $$5x + 6 = 0 \quad \text{or} \quad 3x - 8 = 0.$$ 4. **Solve each equation:** For $5x + 6 = 0$: $$5x = -6$$ $$x = \frac{-6}{5}.$$ For $3x - 8 = 0$: $$3x = 8$$ $$x = \frac{8}{3}.$$ 5. **Final answer:** The two possible values of $x$ are $$x = \frac{-6}{5}, \frac{8}{3}.$$