1. **State the problem:** Find the value(s) of $y$ that satisfy the equation $$(y-1)(3y-4)(2y+5)=0.$$\n\n2. **Formula and rule:** For a product of factors to be zero, at least one of the factors must be zero. This is called the Zero Product Property.\n\n3. **Apply the Zero Product Property:** Set each factor equal to zero:\n$$y-1=0$$\n$$3y-4=0$$\n$$2y+5=0$$\n\n4. **Solve each equation:**\n- For $y-1=0$, add 1 to both sides:\n$$y=1$$\n\n- For $3y-4=0$, add 4 to both sides:\n$$3y=4$$\nNow divide both sides by 3:\n$$y=\frac{\cancel{3}y}{\cancel{3}}=\frac{4}{3}$$\n\n- For $2y+5=0$, subtract 5 from both sides:\n$$2y=-5$$\nNow divide both sides by 2:\n$$y=\frac{\cancel{2}y}{\cancel{2}}=\frac{-5}{2}$$\n\n5. **Final answer:** The values of $y$ are $$y=1, \quad y=\frac{4}{3}, \quad y=\frac{-5}{2}.$$