https://file.aiquickdraw.com/mathbot/file/1725916355125bufzcr5h.jpg

The expression provided in the image is:

\[
x^3 - x^2 + 16x - 16
\]

To factor this completely using grouping, we can proceed with the following steps:

### Step 1: Group the terms
Group the terms in pairs:

\[
(x^3 - x^2) + (16x - 16)
\]

### Step 2: Factor out the greatest common factor (GCF) from each group
- From the first group \((x^3 - x^2)\), the GCF is \(x^2\), so we factor out \(x^2\):

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

- From the second group \((16x - 16)\), the GCF is 16, so we factor out 16:

\[
16(x - 1)
\]

Now the expression looks like this:

\[
x^2(x - 1) + 16(x - 1)
\]

### Step 3: Factor out the common binomial
Both terms contain the factor \((x - 1)\), so factor it out:

\[
(x - 1)(x^2 + 16)
\]

### Final Factored Form:
The fully factored expression is:

\[
(x - 1)(x^2 + 16)
\]

The expression \(x^2 + 16\) cannot be factored further using real numbers since it represents a sum of squares.

---

Would you like any more details on this process or have additional questions?

### Here are 5 questions to explore further:
1. What is the general process for factoring by grouping?
2. Why can't \(x^2 + 16\) be factored further using real numbers?
3. How can you recognize when to use factoring by grouping?
4. What happens when a sum of squares involves complex numbers?
5. Can the same method be applied to higher-degree polynomials?

### Tip:
Always check each group for a common factor before attempting to factor out any binomials.

Learn how to factor the polynomial expression x^3 - x^2 + 16x - 16 completely using factoring by grouping. This solution includes factoring out the greatest common factor (GCF) and recognizing common binomials, suitable for students in Grades 9-10.

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