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:

   = 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: $3a^3 + 5a + 12$.

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!