https://file.aiquickdraw.com/mathbot/file/1726886774736qsvrr182.jpg

Let's go through the expressions and steps visible in the image:

1. **First Expression:**
   \[
   (4m^2 + 6mn + 9n^2)(2m - 3n)
   \]
   This appears to be a multiplication of a quadratic trinomial and a binomial.

   The steps show:
   \[
   (2m + 3n)^3 = 8m^3 - 27n^3
   \]
   This seems to imply an error in the initial calculation since \((2m + 3n)^3\) does not match the original expression. Likely, there's confusion in the expansion.

2. **Second Expression:**
   \[
   (-5a^2 b)(21a^4 b^6 - 6a^3 b + 7a^3 b^4)
   \]
   The steps show a multiplication of \(-5a^2 b\) with each term inside the parentheses:
   \[
   = -105a^6 b^7 - 30a^5 b^2 + 35a^5 b^5
   \]
   which looks correct for this polynomial distribution.

3. **Third Expression:**
   \[
   x y(-2x^2 y + 3xy - 4xy^2) = 2xy(2xy - 7xy^2 + tx^2 y)
   \]
   Some terms are written lightly, but it seems to involve factoring and simplification.

4. **Fourth Expression:**
   \[
   8a^3 - 22a^2 - 5a + 12 \div (4a + 3)
   \]
   This involves polynomial division, though the steps are not fully visible.

5. **Fifth Expression:**
   \[
   (y^3 - 1) \div (y - 1)
   \]
   This is a difference of cubes:
   \[
   y^3 - 1 = (y - 1)(y^2 + y + 1)
   \]
   So the simplified expression should be:
   \[
   y^2 + y + 1
   \]

Would you like detailed steps for any of the problems, or would you like assistance with a specific step or concept from this image?

### Five Follow-up Questions:
1. Would you like a detailed explanation of the expansion in the first problem?
2. Should I help complete the division in the fourth expression?
3. Would you like assistance simplifying the third polynomial expression?
4. Do you need help identifying any missing or faint terms in the image?
5. Would you like further explanations on solving cubic or quadratic expressions?

### Tip:
When multiplying polynomials, distribute each term of one polynomial to every term of the other, taking care of signs and powers.

Various polynomial expressions involving algebraic expansion, polynomial division, and simplification.

This problem involves expanding polynomial expressions, performing polynomial division, and applying algebraic identities like the difference of cubes and squares. Suitable for high school students in Grades 9-11.

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