https://file.aiquickdraw.com/mathbot/file/1726886384210bczeq2st.jpg

Let's go over the steps in the image you've provided:

### 1. First Expression Simplification:
\[
(4m^2 + 6mn + 9n^2)(2m - 3n)
\]
The solution seems to follow factoring and difference of cubes:

- Factor \(4m^2 + 6mn + 9n^2\) as \((2m - 3n)^2\).
- Apply the difference of cubes formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
  
Therefore, the result becomes:
\[
8m^3 - 27n^3
\]

### 2. Polynomial Multiplication:
\[
(-5a^2b)(21a^4b^2 - 6a^3b + 7ab^3 - b^4)
\]

Multiply each term:
- \(-5a^2b \times 21a^4b^2 = -105a^6b^3\),
- \(-5a^2b \times -6a^3b = 30a^5b^2\),
- \(-5a^2b \times 7ab^3 = -35a^3b^4\),
- \(-5a^2b \times -b^4 = 5a^2b^5\).

Result:
\[
-105a^6b^3 + 30a^5b^2 - 35a^3b^4 + 5a^2b^5
\]

### 3. Polynomial Expression:
The third equation starts with \(xy(-2x^2y + 3xy - 4x)\). This will involve distributing \(xy\) across the terms within the parentheses:
\[
xy \times (-2x^2y) = -2x^3y^2,
\]
\[
xy \times 3xy = 3x^2y^2,
\]
\[
xy \times (-4x) = -4x^2y.
\]

Result:
\[
-2x^3y^2 + 3x^2y^2 - 4x^2y
\]

### Next Steps:
Would you like a detailed breakdown of the fourth and remaining expressions? Feel free to ask!

Here are 5 related questions for deeper practice:
1. Can you explain how to factor a sum or difference of cubes?
2. What are the key steps for distributing terms in polynomial multiplication?
3. How do you handle terms with multiple variables when multiplying polynomials?
4. What is the difference between factoring trinomials and factoring cubes?
5. Can you generalize how to apply the distributive property in polynomial expressions?

**Tip:** Always remember to group like terms and simplify wherever possible to avoid mistakes during complex polynomial multiplications.

Simplify and solve the following expressions: (4m^2 + 6mn + 9n^2)(2m - 3n), (-5a^2b)(21a^4b^2 - 6a^3b + 7ab^3 - b^4), and xy(-2x^2y + 3xy - 4x).

Learn how to simplify polynomial expressions through multiplication and factoring cubes. This lesson includes step-by-step solutions to complex algebraic expressions involving variables, exponents, and polynomial terms. Ideal for students in grades 9-12.

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