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The given expression is \( x^3 - 6x^2 + x - 6 \), and the task is to factor it completely by grouping.

### Step-by-step factoring:
1. Group the terms: 
   \[
   (x^3 - 6x^2) + (x - 6)
   \]
2. Factor out the common factor from each group:
   \[
   x^2(x - 6) + 1(x - 6)
   \]
3. Now, factor out the common binomial \( (x - 6) \):
   \[
   (x^2 + 1)(x - 6)
   \]

Thus, the factored form of the expression is \( (x^2 + 1)(x - 6) \).

### Correct Answer:
The correct option is:
\[
\boxed{(x^2 + 1)(x - 6)}
\]

Would you like more details or have any other questions?

Here are 5 related questions:
1. How do you factor expressions that involve four terms?
2. What is the process of grouping in polynomial factoring?
3. How would you factor a cubic expression without grouping?
4. Can every cubic expression be factored by grouping?
5. What is the significance of finding common factors in factoring?

**Tip**: When factoring by grouping, always check if the terms in both groups have a common binomial factor that can be factored out.

Factor completely by grouping: x^3 - 6x^2 + x - 6

Learn how to factor the cubic polynomial x^3 - 6x^2 + x - 6 using the grouping method. This solution includes detailed steps such as factoring by grouping and extracting common factors, suitable for high school algebra students.

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