Let's solve the given expression:

$\text{Simplify: } \left(-x + 2 + 4x^2 \right) - \left(-x - 6 - 6x^3 \right)$

Step 1: Remove the parentheses

To simplify the expression, first remove the parentheses by distributing the subtraction across the second group:

$- x + 2 + 4x^2 - (-x - 6 - 6x^3)$

Distribute the negative sign to each term in the second group:

$- x + 2 + 4x^2 + x + 6 + 6x^3$

Step 2: Combine like terms

Now, combine the like terms:

• Combine the $x$ terms: $-x + x = 0$
• The constant terms: $2 + 6 = 8$
• The $x^2$ term remains $4x^2$
• The $x^3$ term is $6x^3$

Thus, the simplified expression is:

$6x^3 + 4x^2 + 8$

This is the final simplified expression. Let me know if you have any questions or if you need more details.

Here are some related questions you might find interesting:

1. How do you factor a polynomial like $6x^3 + 4x^2 + 8$?
2. What are the steps to differentiate the polynomial $6x^3 + 4x^2 + 8$?
3. How would you solve $6x^3 + 4x^2 + 8 = 0$?
4. Can you graph the polynomial $6x^3 + 4x^2 + 8$? What are its key features?
5. How do the degrees of each term affect the shape of the graph of $6x^3 + 4x^2 + 8$?

Tip: When simplifying expressions with multiple terms, it's crucial to carefully distribute negative signs to ensure accuracy.