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.