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The given expression is:

\[
(-t^3 - 5t^2 - 6t + 3) + (4t^4 + 6t^3 + 6t^2 - 5t + 2)
\]

We need to combine like terms and simplify.

### Step 1: Grouping like terms
\[
4t^4
\]
For \(t^3\):
\[
-t^3 + 6t^3 = 5t^3
\]
For \(t^2\):
\[
-5t^2 + 6t^2 = t^2
\]
For \(t\):
\[
-6t - 5t = -11t
\]
Constant terms:
\[
3 + 2 = 5
\]

### Step 2: Combine and write the simplified expression

\[
4t^4 + 5t^3 + t^2 - 11t + 5
\]

This is the simplified form.

Would you like more details on any part of the solution? Here are some additional related questions to explore:

1. How do you combine like terms in polynomial expressions?
2. What are the different types of terms in polynomials (e.g., monomials, binomials)?
3. How do you determine the degree of a polynomial?
4. How can you factor polynomials after simplifying them?
5. How do you use the distributive property in polynomial operations?

**Tip:** When simplifying polynomials, always make sure to organize terms by their degree for easier grouping!

Perform the operation: (-t^3 - 5t^2 - 6t + 3) + (4t^4 + 6t^3 + 6t^2 - 5t + 2) and combine like terms. Simplify your answer as much as possible.

Learn how to simplify the polynomial expression (-t^3 - 5t^2 - 6t + 3) + (4t^4 + 6t^3 + 6t^2 - 5t + 2) by combining like terms. This solution covers essential algebraic concepts, including the commutative and associative properties, and is suitable for students in Grades 9-11.

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