https://file.aiquickdraw.com/mathbot/file/17296233553978z4l5m3x.jpg

Let's break down the problem in the image. We are tasked with subtracting two polynomials:

\[
\left(a^3 + a^2 + 2a \right) - \left( -2a^3 + a^2 - 3a - 12 \right)
\]

### Step-by-Step Solution:

1. **Distribute the negative sign to the second polynomial**:
   \[
   \left(a^3 + a^2 + 2a\right) - \left(-2a^3 + a^2 - 3a - 12\right)
   = a^3 + a^2 + 2a + 2a^3 - a^2 + 3a + 12
   \]

2. **Combine like terms**:
   - For \(a^3\): \(a^3 + 2a^3 = 3a^3\)
   - For \(a^2\): \(a^2 - a^2 = 0\)
   - For \(a\): \(2a + 3a = 5a\)
   - For constants: The constant term is just \(12\).

3. **Final expression**:
   \[
   3a^3 + 5a + 12
   \]

Thus, the correct answer is **Option B**: \(3a^3 + 5a + 12\).

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Would you like more details or have any questions?

### Here are some additional questions for practice:

1. How do you distribute negative signs in polynomials?
2. What happens to terms with the same degree in subtraction or addition?
3. How do you combine terms like \(a^3\) and \(a^2\) in polynomials?
4. What is the degree of the polynomial after subtraction?
5. What role do constants play in polynomial operations?

**Tip**: Always group like terms after distributing to simplify complex expressions!

Subtract: (a^3 + a^2 + 2a) - (-2a^3 + a^2 - 3a - 12)

Learn how to subtract two polynomials: (a^3 + a^2 + 2a) - (-2a^3 + a^2 - 3a - 12). This problem explains the process of distributing the negative sign, combining like terms, and simplifying the expression. Suitable for students in Grades 9-11.

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