https://file.aiquickdraw.com/mathbot/file/17268517474307mfooniz.jpg

The expression in the image is:

\[
2x^5 - 4x^4 - 6x + x^5 + x^4 - 3x^3 - 3x^2
\]

To find the HCF (Highest Common Factor), let's break down each term and factor out the greatest common factor from the entire expression.

### Step-by-step factoring:
1. **Combine like terms**:
   - \( 2x^5 + x^5 = 3x^5 \)
   - \( -4x^4 + x^4 = -3x^4 \)

   The simplified expression becomes:
   \[
   3x^5 - 3x^4 - 3x^3 - 3x^2 - 6x
   \]

2. **Factor out the common term**:
   Each term contains a factor of \( 3x \), so we can factor out \( 3x \):

   \[
   3x(x^4 - x^3 - x^2 - x - 2)
   \]

Thus, the expression factored by its HCF is:

\[
3x(x^4 - x^3 - x^2 - x - 2)
\]

Let me know if you'd like further details or clarifications!

### Relative questions:
1. How do you find the greatest common factor in polynomials?
2. What are the next steps after factoring out the HCF?
3. Can this expression be factored further beyond the HCF?
4. How does combining like terms simplify polynomial expressions?
5. What role does factoring play in solving polynomial equations?

### Tip:
When factoring, always check for the highest degree terms and the greatest numerical factor common to all terms for a smoother process.

Learn how to factor the polynomial 2x^5 - 4x^4 - 6x + x^5 + x^4 - 3x^3 - 3x^2 by taking out the HCF. This detailed step-by-step solution simplifies the polynomial by combining like terms and factoring out the greatest common factor. Suitable for students in Grades 9-12.

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