Solve the polynomial equation $16y^2 = 32y^3 + 2y$ using factoring and the Zero Product Property.

Step 1 — Write in standard form. Subtract $16y^2$ from both sides so that one side equals zero. Rearrange in descending order: $32y^3 - 16y^2 + 2y = 0$

Step 2 — Factor the greatest common factor. All terms share a common factor of $2y$. Factor it out: $2y(16y^2 - 8y + 1) = 0$

Step 3 — Factor the trinomial. The resulting trinomial is a perfect square trinomial: $2y(4y - 1)(4y - 1) = 0$

Step 4 — Apply the Zero Product Property. Set each variable factor equal to zero: $2y = 0 \quad \text{or} \quad 4y - 1 = 0 \quad \text{or} \quad 4y - 1 = 0$

Step 5 — Solve each equation: $y = 0 \quad \text{or} \quad y = \frac{1}{4} \quad \text{or} \quad y = \frac{1}{4}$

The distinct solutions are $y = 0$ and $y = \frac{1}{4}$.