To simplify the given expression \((-3c^4 - 5c^2 - 7c) + (4c^4 + 8c^3 + 2c^2)\), we'll combine like terms. Let's break it down step-by-step:

### Step 1: Identify Like Terms
Like terms are terms that have the same variable raised to the same power.

1. \(c^4\) terms: \(-3c^4\) and \(4c^4\)
2. \(c^3\) terms: \(8c^3\) (only one term)
3. \(c^2\) terms: \(-5c^2\) and \(2c^2\)
4. \(c\) terms: \(-7c\) (only one term)

### Step 2: Combine Like Terms
Now we combine the coefficients of like terms:

1. \(c^4\) terms: \(-3c^4 + 4c^4 = 1c^4\)
2. \(c^3\) term remains: \(8c^3\)
3. \(c^2\) terms: \(-5c^2 + 2c^2 = -3c^2\)
4. \(c\) term remains: \(-7c\)

### Step 3: Write the Simplified Expression
Combining all terms, we get:

\[
c^4 + 8c^3 - 3c^2 - 7c
\]

### Final Answer
\[
c^4 + 8c^3 - 3c^2 - 7c
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you add and subtract polynomials?
2. What are like terms, and why is it important to combine them?
3. How would you multiply two polynomials?
4. What happens when there are no like terms in a polynomial addition or subtraction?
5. How do you handle exponents when multiplying and dividing polynomials?

**Tip:** When combining like terms, always pay attention to the signs of the coefficients.

Learn how to simplify the polynomial expression -3c^4 - 5c^2 - 7c + 4c^4 + 8c^3 + 2c^2 step-by-step. This problem involves combining like terms in polynomials. Suitable for students in Grades 7-9.

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