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The problem asks to expand and simplify the expression \((2a^2 + b)(2a - 3b^2)\).

Let's solve this step-by-step using the distributive property (also called FOIL method for binomials):

### Step 1: Distribute each term in the first bracket to each term in the second bracket.

\[
(2a^2 + b)(2a - 3b^2)
\]

This will give us:

\[
= 2a^2 \cdot 2a + 2a^2 \cdot (-3b^2) + b \cdot 2a + b \cdot (-3b^2)
\]

### Step 2: Multiply the terms.

1. \(2a^2 \cdot 2a = 4a^3\)
2. \(2a^2 \cdot (-3b^2) = -6a^2b^2\)
3. \(b \cdot 2a = 2ab\)
4. \(b \cdot (-3b^2) = -3b^3\)

### Step 3: Combine all the terms together.

\[
4a^3 - 6a^2b^2 + 2ab - 3b^3
\]

Thus, the expanded and simplified form of \((2a^2 + b)(2a - 3b^2)\) is:

\[
\boxed{4a^3 - 6a^2b^2 + 2ab - 3b^3}
\]

Would you like further clarification or more details on this solution?

### Related Questions:
1. How would you expand and simplify \((x + 2)(x^2 - 4)\)?
2. Can you expand \((3a + 4b)(5a - 6b)\) and simplify it?
3. What is the simplified result of \((a - b)(a^2 + ab + b^2)\)?
4. Can you explain how to factor the expression \(x^2 + 5x + 6\)?
5. How does the FOIL method work for binomial expansions?

### Tip:
When expanding expressions, always carefully apply the distributive property and keep track of the signs (positive/negative) to avoid errors.

Expand and simplify (2a^2 + b)(2a - 3b^2)

Learn how to expand and simplify the expression (2a^2 + b)(2a - 3b^2). This step-by-step solution covers key algebraic concepts like polynomial expansion and binomial multiplication. Ideal for students in Grades 9-11.

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