Expand and simplify the expression (3*a* − 2*b*)(6*a* − *b*) + 4*a*(9*a* + 4*b*).

Let's expand and simplify the expression:

\[
(3a - 2b)(6a - b) + 4a(9a + 4b)
\]

### Step 1: Expand each part of the expression.

1. **Expand \((3a - 2b)(6a - b)\) using the distributive property:**
   \[
   (3a)(6a) + (3a)(-b) + (-2b)(6a) + (-2b)(-b)
   \]
   \[
   = 18a^2 - 3ab - 12ab + 2b^2
   \]
   Combine the like terms:
   \[
   18a^2 - 15ab + 2b^2
   \]

2. **Expand \(4a(9a + 4b)\):**
   \[
   4a(9a) + 4a(4b) = 36a^2 + 16ab
   \]

### Step 2: Combine the expanded parts.

\[
(18a^2 - 15ab + 2b^2) + (36a^2 + 16ab)
\]

Combine like terms:

- For \(a^2\) terms:
  \[
  18a^2 + 36a^2 = 54a^2
  \]

- For \(ab\) terms:
  \[
  -15ab + 16ab = 1ab \quad \text{(or simply \(ab\))}
  \]

- For \(b^2\) terms:
  \[
  2b^2
  \]

### Final Simplified Expression:

\[
54a^2 + ab + 2b^2
\]

This is the expanded and simplified form of the given expression.

Would you like more details or have any questions about this process?

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Here are 5 related questions you might find interesting:

1. How would the expansion change if the sign of \(b\) were reversed in the original expression?
2. What would happen if the coefficients of \(a\) and \(b\) were doubled in the original expression?
3. How do you factor the expression \(54a^2 + ab + 2b^2\)?
4. What is the geometric interpretation of the simplified expression?
5. How would the solution change if the powers of \(a\) and \(b\) were different in the original expression?

**Tip:** When expanding and simplifying expressions, it helps to carefully track each term to avoid mistakes in combining like terms.

Learn how to expand and simplify the expression (3a - 2b)(6a - b) + 4a(9a + 4b) step-by-step. This problem covers algebraic concepts like the distributive property and combining like terms. Suitable for high school students (Grades 9-12).

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