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To solve this problem, we need to expand the given polynomial expression:

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

### Step 1: Expand \((x - 2)^2\)
First, expand the square of \((x - 2)\):
\[
(x - 2)^2 = (x - 2)(x - 2) = x^2 - 4x + 4
\]

### Step 2: Multiply by \(x\)
Now, multiply the result by \(x\):
\[
x(x^2 - 4x + 4) = x^3 - 4x^2 + 4x
\]

### Step 3: Multiply by \((x + 4)\)
Next, distribute \((x + 4)\) across \((x^3 - 4x^2 + 4x)\):
\[
(x^3 - 4x^2 + 4x)(x + 4)
\]

Using the distributive property:
\[
x^3(x + 4) = x^4 + 4x^3
\]
\[
-4x^2(x + 4) = -4x^3 - 16x^2
\]
\[
4x(x + 4) = 4x^2 + 16x
\]

### Step 4: Combine like terms
Now, add all the terms:
\[
x^4 + 4x^3 - 4x^3 - 16x^2 + 4x^2 + 16x
\]

Simplify:
\[
x^4 + (4x^3 - 4x^3) + (-16x^2 + 4x^2) + 16x
\]
\[
x^4 - 12x^2 + 16x
\]

### Final Answer:
The expanded polynomial is:

\[
x^4 - 12x^2 + 16x
\]

Thus, the correct answer is:

\[
\boxed{x^4 - 12x^2 + 16x}
\]

Let me know if you need more details or explanations!

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Here are 5 related questions to expand your understanding:
1. How do you expand the cube of a binomial expression, such as \((x - 2)^3\)?
2. What is the general formula for expanding \((a - b)^n\) using the binomial theorem?
3. How do you factor the polynomial \(x^4 - 12x^2 + 16x\)?
4. How do you solve for the roots of a quadratic polynomial like \(x^2 - 4x + 4\)?
5. How can you verify the correctness of a polynomial expansion?

**Tip:** When expanding polynomials, always distribute terms carefully and combine like terms to avoid errors.

When expanded, the polynomial x(x - 2)^2(x + 4) is equal to which of the following?

Learn how to expand the polynomial expression x(x - 2)^2(x + 4) using algebraic concepts such as the distributive property and binomial expansion. This detailed step-by-step guide simplifies the expression and combines like terms. Suitable for students in Grades 9-11.

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