What is the following product? Assume \( y \geq 0 \) \( 3 \sqrt{10}\left(y^{2} \sqrt{4}+\sqrt{8 y}\right) \) \( 6 y^{2} \sqrt{10}+12 \sqrt{5 y} \) \( 6 \sqrt{10}+12 \sqrt{5 y} \) \( 6 y^{2} \sqrt{10}+4 \sqrt{5 y} \) \( 3 y^{2} \sqrt{10}+12 \sqrt{5 y} \)

Let’s break down the original expression step by step to see what we can determine about it! First, simplifying what’s inside the parentheses: 1. The term \( \sqrt{4} \) simplifies to \( 2 \). 2. The term \( \sqrt{8y} \) simplifies to \( 2 \sqrt{2y} \). So the expression \( y^{2} \sqrt{4} + \sqrt{8y} \) becomes: \[ y^{2} \cdot 2 + 2 \sqrt{2y} = 2y^{2} + 2\sqrt{2y} \] Now, we can factor out a 2: \[ 2(y^{2} + \sqrt{2y}) \] Now we can go back to the original expression: \[ 3 \sqrt{10} \cdot 2(y^{2} + \sqrt{2y}) = 6\sqrt{10}(y^{2} + \sqrt{2y}) \] Distributing \( 6\sqrt{10} \) gives us: \[ 6\sqrt{10}y^{2} + 6\sqrt{20y} \] Since \( \sqrt{20} \) simplifies to \( 2\sqrt{5} \): \[ 6\sqrt{10}y^{2} + 12\sqrt{5y} \] The resulting expression is: \[ 6 y^{2} \sqrt{10} + 12 \sqrt{5 y} \] Now, when looking at the options: 1. \( 6 y^{2} \sqrt{10}+12 \sqrt{5 y} \) is a direct match. 2. The other options are incorrect representations based on our simplification. Thus, the correct product is \( 6 y^{2} \sqrt{10} + 12 \sqrt{5y} \). Remember, when simplifying expressions like this, always take it step by step to avoid missing any foundational components!