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Let's solve the equation \( 2x^3 - 12x^2 - 54x \) by factoring it completely.

### Step 1: Factor out the greatest common factor (GCF)
All the terms in \( 2x^3 - 12x^2 - 54x \) share a common factor of \( 2x \). So, let's factor that out:

\[
2x(x^2 - 6x - 27)
\]

### Step 2: Factor the quadratic \( x^2 - 6x - 27 \)
We need to factor \( x^2 - 6x - 27 \). To do this, find two numbers that multiply to give \( -27 \) and add to give \( -6 \). The numbers are \( -9 \) and \( 3 \), since:

\[
-9 \times 3 = -27 \quad \text{and} \quad -9 + 3 = -6
\]

Thus, we can factor \( x^2 - 6x - 27 \) as \( (x - 9)(x + 3) \).

### Step 3: Put everything together
Now, rewrite the expression as:

\[
2x(x - 9)(x + 3)
\]

### Final Answer:
The fully factored form is \( 2x(x - 9)(x + 3) \), which corresponds to **option b**.

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Would you like further details or explanations? Here are some related questions to deepen your understanding:

1. How do you factor a quadratic when it doesn't factor easily with integers?
2. What is the difference between factoring by grouping and using the quadratic formula?
3. Can all cubic polynomials be factored this way, or are there special cases?
4. How can you verify your factored expression is correct?
5. What role does the GCF play in simplifying polynomial expressions?

**Tip:** Always check your factorization by multiplying the factors back together to ensure you get the original expression.

Learn how to factor the cubic polynomial 2x^3 - 12x^2 - 54x using the GCF and quadratic factoring techniques. This step-by-step solution explains each stage of the process and is suitable for high school students (Grades 9-12).

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