https://file.aiquickdraw.com/mathbot/file/1727276146525np3jbg35.jpg

The problem in the image is asking to add two polynomials:

\[
(-x - x^3 - 5x^2) + (-6x^3 - 7x^2 - x)
\]

Let's break down the steps to add these polynomials:

### Step 1: Combine Like Terms
1. **Terms with \(x^3\)**: 
   \[
   -x^3 + (-6x^3) = -7x^3
   \]

2. **Terms with \(x^2\)**: 
   \[
   -5x^2 + (-7x^2) = -12x^2
   \]

3. **Terms with \(x\)**: 
   \[
   -x + (-x) = -2x
   \]

### Step 2: Write the Result
The sum of the polynomials is:
\[
-7x^3 - 12x^2 - 2x
\]

### Step 3: Check the Choices
Looking at the options in the image:
- A) \(-x^3 - 9x^2 - 7x^2 - 2x\) → Doesn't match
- B) \(-x^3 - 16x^2 - 5x^2 - 2x\) → Doesn't match
- C) \(-x^3 - 9x^2 - 8x^2 - 2x\) → Doesn't match
- D) \(-7x^3 - 12x^2 - 2x\) → This matches our result.

Thus, the correct answer is **D**.

Would you like any further details or explanations?

Here are 5 related questions:
1. How do you add polynomials with more variables?
2. How can you simplify polynomials with different degrees?
3. What are like terms, and why are they important in polynomial operations?
4. How do you subtract polynomials, and how is it different from addition?
5. What are some real-life applications of polynomials in different fields?

**Tip**: Always line up terms by their degree to ensure you are combining like terms correctly!

Add the polynomials: (-x - x^3 - 5x^2) + (-6x^3 - 7x^2 - x)

Learn how to add the polynomials (-x - x^3 - 5x^2) and (-6x^3 - 7x^2 - x). This step-by-step solution explains combining like terms, resulting in the polynomial: -7x^3 - 12x^2 - 2x. Ideal for students in Grades 8-10.

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