Factor completely. \[ 3 y^{4}+3 y^{3}-9 y^{2}-9 y \]

To factor the expression \( 3 y^{4} + 3 y^{3} - 9 y^{2} - 9 y \), we can start by factoring out the greatest common factor, which is \( 3y \): \[ 3y(y^{3} + y^{2} - 3y - 3) \] Next, we focus on factoring the polynomial \( y^{3} + y^{2} - 3y - 3 \). We can apply factoring by grouping: 1. Group the terms: \( (y^{3} + y^{2}) + (-3y - 3) \). 2. Factor out common factors from each group: - From the first group, we factor out \( y^{2} \): \( y^{2}(y + 1) \). - From the second group, we factor out \(-3\): \(-3(y + 1)\). Now, we can rewrite the expression as: \[ 3y\left(y^{2}(y + 1) - 3(y + 1)\right) \] Notice that \( (y + 1) \) is a common factor: \[ 3y(y + 1)\left(y^{2} - 3\right) \] The term \( y^{2} - 3 \) is a difference of squares. We can factor that further: \[ y^{2} - 3 = (y - \sqrt{3})(y + \sqrt{3}) \] Putting it all together, we have the complete factorization: \[ 3y(y + 1)(y - \sqrt{3})(y + \sqrt{3}) \]