To find the correct polynomial product, we'll use the distributive property (or the area model, as hinted by the grid in the image) to multiply the binomial $(2y + 3)$ by the trinomial $(3y^2 + 4y + 5)$.

Let's fill in the grid by multiplying each term in the binomial by each term in the trinomial:

Step 1: Distribute $2y$ across the trinomial:

• $2y \times 3y^2 = 6y^3$
• $2y \times 4y = 8y^2$
• $2y \times 5 = 10y$

Step 2: Distribute $3$ across the trinomial:

• $3 \times 3y^2 = 9y^2$
• $3 \times 4y = 12y$
• $3 \times 5 = 15$

Step 3: Add all these products together:

$6y^3 + 8y^2 + 10y + 9y^2 + 12y + 15$

Step 4: Combine like terms:

• $y^3$ terms: $6y^3$
• $y^2$ terms: $8y^2 + 9y^2 = 17y^2$
• $y$ terms: $10y + 12y = 22y$
• Constant term: $15$

Final Simplified Expression:

$6y^3 + 17y^2 + 22y + 15$

Step 5: Match with the provided options:

The correct polynomial is: $\boxed{6y^3 + 17y^2 + 22y + 15}$

This matches the second option.

Do you want further details or have any questions?

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Here are 5 related questions you might find useful:

1. How do you expand and simplify the product of two polynomials?
2. What are the benefits of using the area model (or grid method) in polynomial multiplication?
3. How do you identify like terms in a polynomial expression?
4. How can polynomial multiplication be applied in solving real-world problems?
5. How would you factor a polynomial like $6y^3 + 17y^2 + 22y + 15$?

Tip: When multiplying polynomials, organizing your work in a grid or table can help ensure that all terms are correctly accounted for and combined.