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The problem in the image involves multiplying two polynomials and simplifying the result. Here's the given expression:

\[
(3y^3 - 4y + 5)(3y^2 - 3y + 6)
\]

We will multiply each term in the first polynomial by each term in the second polynomial using the distributive property and then combine like terms. Let's expand the product step-by-step:

### Step 1: Multiply each term in the first polynomial by each term in the second polynomial.
1. \( 3y^3 \cdot 3y^2 = 9y^5 \)
2. \( 3y^3 \cdot (-3y) = -9y^4 \)
3. \( 3y^3 \cdot 6 = 18y^3 \)

4. \( -4y \cdot 3y^2 = -12y^3 \)
5. \( -4y \cdot (-3y) = 12y^2 \)
6. \( -4y \cdot 6 = -24y \)

7. \( 5 \cdot 3y^2 = 15y^2 \)
8. \( 5 \cdot (-3y) = -15y \)
9. \( 5 \cdot 6 = 30 \)

### Step 2: Combine like terms.
Now let's collect the terms with the same powers of \( y \).

- \( y^5 \) term: \( 9y^5 \)
- \( y^4 \) term: \( -9y^4 \)
- \( y^3 \) terms: \( 18y^3 - 12y^3 = 6y^3 \)
- \( y^2 \) terms: \( 12y^2 + 15y^2 = 27y^2 \)
- \( y \) terms: \( -24y - 15y = -39y \)
- Constant term: \( 30 \)

### Final result:
\[
9y^5 - 9y^4 + 6y^3 + 27y^2 - 39y + 30
\]

This is the simplified expression.

Would you like further details or have any questions?

Here are some related questions you might find useful:
1. How do you handle the multiplication of polynomials with more than two terms?
2. What are the common mistakes to avoid when combining like terms?
3. How does the distributive property apply to polynomial multiplication?
4. Can you explain the significance of the degree of the polynomial in the final result?
5. How would you multiply polynomials if one of them had more variables?

**Tip:** Always arrange the terms in decreasing order of exponents when simplifying polynomials to avoid confusion while combining like terms.

Learn how to perform polynomial multiplication by expanding (3y^3 - 4y + 5)(3y^2 - 3y + 6). This step-by-step guide explains how to apply the distributive property and combine like terms. Suitable for students in Grades 9-12.

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