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The problem in the image asks you to divide the polynomial \( 21x^6 + 63x^4 \) by \(-7x\).

Let's go through the steps to solve this:

1. **Write the expression**:  
   \[
   \frac{21x^6 + 63x^4}{-7x}
   \]

2. **Divide each term by \(-7x\)**:
   - For the first term:  
     \[
     \frac{21x^6}{-7x} = -3x^{6-1} = -3x^5
     \]
   - For the second term:  
     \[
     \frac{63x^4}{-7x} = -9x^{4-1} = -9x^3
     \]

3. **Combine the results**:  
   \[
   \frac{21x^6 + 63x^4}{-7x} = -3x^5 - 9x^3
   \]

So, the simplified expression after dividing is:
\[
-3x^5 - 9x^3
\]

Would you like more details or have any other questions?

Here are five related questions to further your understanding:
1. How would the solution change if the divisor was \(7x\) instead of \(-7x\)?
2. What happens if you divide the polynomial \(21x^6 + 63x^4\) by \(7x^2\)?
3. Can you factor out the greatest common factor from the polynomial \(21x^6 + 63x^4\) before dividing?
4. What is the result of dividing a polynomial by a monomial with a negative coefficient?
5. How would you check your answer to ensure it's correct?

**Tip**: When dividing polynomials, always simplify each term individually and ensure you account for negative signs correctly.

Divide the following polynomial by -7x: 21x^6 + 63x^4

Learn how to divide the polynomial 21x^6 + 63x^4 by -7x with a step-by-step solution. This problem explores key algebraic concepts and polynomial division, suitable for students in Grades 9-12.

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