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The given expression is:

\[
x^4 - 9x^2 + 6x^3 - 54x + 8x^2 - 72
\]

Let's factor this by grouping. First, rearrange the terms into two groups that might allow factoring:

\[
(x^4 + 6x^3 + 8x^2) - (9x^2 + 54x + 72)
\]

### Step 1: Factor each group.
- For the first group \((x^4 + 6x^3 + 8x^2)\), we can factor out \(x^2\):
  \[
  x^2(x^2 + 6x + 8)
  \]
  The quadratic \(x^2 + 6x + 8\) factors as \((x + 4)(x + 2)\), so we get:
  \[
  x^2(x + 4)(x + 2)
  \]

- For the second group \((9x^2 + 54x + 72)\), we can factor out 9:
  \[
  9(x^2 + 6x + 8)
  \]
  The quadratic \(x^2 + 6x + 8\) is the same, so we have:
  \[
  9(x + 4)(x + 2)
  \]

### Step 2: Factor out the common term \((x + 4)(x + 2)\) from both groups:
\[
(x^2 - 9)(x + 4)(x + 2)
\]

### Step 3: Notice that \(x^2 - 9\) is a difference of squares:
\[
x^2 - 9 = (x - 3)(x + 3)
\]

### Final factored form:
\[
(x - 3)(x + 3)(x + 4)(x + 2)
\]

Thus, the fully factored form of the expression is:

\[
(x - 3)(x + 3)(x + 4)(x + 2)
\]

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:
1. How can you identify when to use grouping for factoring?
2. What are other examples of factoring differences of squares?
3. What strategies can help in factoring complex polynomials more efficiently?
4. How does factoring relate to solving quadratic equations?
5. Can you apply the factoring technique used here to factor higher-degree polynomials?

**Tip:** Always check for common factors before attempting more advanced factoring methods. This can simplify the process significantly.

Factor the following expression completely: x^4 - 9x^2 + 6x^3 - 54x + 8x^2 - 72

Learn how to factor the polynomial x^4 - 9x^2 + 6x^3 - 54x + 8x^2 - 72 by grouping terms and applying the difference of squares method. Suitable for high school students in Grades 9-12, this detailed solution breaks down each step clearly.

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