https://file.aiquickdraw.com/mathbot/file/1726746427092e4n0v4gd.jpg

The expression in the image is:

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

Let's factorize this polynomial step by step.

### Step 1: Check for common factors
There are no common factors among all terms.

### Step 2: Factor by grouping
Group the terms:

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

Now, factor each group:

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

### Step 3: Factor the common binomial
Notice that \((-x + 1)\) is the same as \((x - 1)\), so we rewrite:

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

Now, factor out the common binomial \((x - 1)\):

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

### Step 4: Simplify the remaining factor
\[
(x - 1)(-x^2 - 2)
\]

This is the factorized form of the given expression.

Would you like further clarification or more details on the steps?

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Here are five related questions you might explore:

1. What is the significance of factoring a cubic polynomial in solving equations?
2. How would the graph of this polynomial look based on its factorization?
3. How can synthetic division be used to check the factorization?
4. Can this cubic expression have real roots? How would you find them?
5. How can factoring by grouping be applied to other cubic polynomials?

**Tip**: Always double-check your factorizations by expanding the factors back to the original expression to confirm accuracy.

Learn how to factor the cubic polynomial -x^3 + x^2 - 2x + 2 with a detailed step-by-step solution. This problem involves polynomial factorization techniques, grouping, and factoring common binomials. Suitable for high school students in Grades 9-12.

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