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Let's factor the given expression: 

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

### Step 1: Factor out the greatest common factor (GCF)
The common factor in all terms is \( -x \). So, we can factor out \( -x \) from the entire expression:

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

### Step 2: Rearrange terms
Now, rearrange the terms inside the parentheses to have them in descending order:

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

### Step 3: Factor further (if possible)
We now need to check if the expression inside the parentheses \( x^4 + x^2 - 2 \) can be factored. This is a quadratic in terms of \( x^2 \). Let \( y = x^2 \), so the expression becomes:

\[
y^2 + y - 2
\]

This can be factored as:

\[
(y - 1)(y + 2)
\]

### Step 4: Substitute back \( y = x^2 \)
Now, replace \( y \) with \( x^2 \):

\[
(x^2 - 1)(x^2 + 2)
\]

### Step 5: Factor \( x^2 - 1 \)
Notice that \( x^2 - 1 \) is a difference of squares, so it can be factored as:

\[
(x - 1)(x + 1)
\]

### Final factored form:
Thus, the fully factored expression is:

\[
-x(x - 1)(x + 1)(x^2 + 2)
\]

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Would you like further details or clarifications on any step?

Here are 5 related questions to expand on this:

1. How do you factor a difference of squares like \(x^2 - 1\)?
2. What methods can be used to factor quadratic expressions?
3. How can you check if a polynomial is factored completely?
4. What is the role of substitution when factoring polynomials?
5. How can you recognize the greatest common factor (GCF) in a polynomial?

**Tip:** When factoring, always check if you can factor out a GCF first before moving on to more complex factoring techniques!

Factor the expression completely: -x^3 + 2x - x^5

Learn how to factor the polynomial expression -x^3 + 2x - x^5 using common factor extraction, substitution, and difference of squares. This problem involves key algebraic concepts and techniques suitable for students in Grades 9-12.

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