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The given expression is:

\[
x^3 + 6x^2 - 9x - 54
\]

We will attempt to factor it step by step.

### Step 1: Group terms
Group the terms in pairs to try factoring by grouping:

\[
(x^3 + 6x^2) - (9x + 54)
\]

### Step 2: Factor out the greatest common factor (GCF) from each group
- From \(x^3 + 6x^2\), factor out \(x^2\):
  \[
  x^2(x + 6)
  \]
- From \(-9x - 54\), factor out \(-9\):
  \[
  -9(x + 6)
  \]

So, the expression becomes:
\[
x^2(x + 6) - 9(x + 6)
\]

### Step 3: Factor out the common binomial
Both terms have a common factor of \((x + 6)\):
\[
(x + 6)(x^2 - 9)
\]

### Step 4: Factor the difference of squares
The second factor, \(x^2 - 9\), is a difference of squares, which can be factored as:
\[
x^2 - 9 = (x - 3)(x + 3)
\]

### Final factored form
Thus, the complete factorization is:
\[
(x + 6)(x - 3)(x + 3)
\]

Therefore, the correct choice is A, with the answer:

\[
x^3 + 6x^2 - 9x - 54 = (x + 6)(x - 3)(x + 3)
\]

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Do you want more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you factor polynomials using the grouping method?
2. What is the difference of squares, and how do you recognize it?
3. What is the greatest common factor (GCF), and how do you factor it from polynomials?
4. How do you know when a polynomial is prime?
5. Can you apply synthetic division to factor higher-degree polynomials?

**Tip:** Always check if you can factor out the greatest common factor (GCF) before attempting more advanced methods.

Factor the expression completely, or state that the polynomial is prime: x^3 + 6x^2 - 9x - 54.

Learn how to factor the cubic polynomial x^3 + 6x^2 - 9x - 54 step by step using factoring by grouping and the difference of squares. This detailed guide covers important algebraic techniques and is suitable for students in Grades 9-11.

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