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Let's analyze the given expression and factor it step-by-step.

The expression is:

\[
4x^4y - 8x^3y - 32x^2y
\]

### Step 1: Factor out common terms
All terms share a common factor of \( 4x^2y \). Let's factor this out:

\[
4x^2y(x^2 - 2x - 8)
\]

### Step 2: Factor the quadratic expression
Now, we need to factor the quadratic expression \( x^2 - 2x - 8 \). To do this, we find two numbers that multiply to \(-8\) and add to \(-2\). These numbers are \( -4 \) and \( 2 \).

So, the quadratic factors as:

\[
x^2 - 2x - 8 = (x - 4)(x + 2)
\]

### Step 3: Substitute the factored form
Now, substitute the factored quadratic back into the original expression:

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

This matches option **D**.

### Answer:
The correct answer is **D**.

Would you like more details or have any questions? Here are five related questions for practice:

1. How do you factor a quadratic trinomial in general?
2. What is the distributive property and how does it apply to factoring?
3. Can you explain how to factor expressions involving higher powers of \( x \)?
4. What are the conditions under which you can factor out common terms in polynomials?
5. How does factoring relate to solving polynomial equations?

**Tip:** Always check for the greatest common factor (GCF) before attempting to factor quadratic or higher-order polynomials!

Which expression is equivalent to 4x^4y - 8x^3y - 32x^2y?

Learn how to factor the expression 4x^4y - 8x^3y - 32x^2y in this detailed guide. We apply algebraic concepts such as factoring out the greatest common factor and factoring quadratic expressions. Ideal for students in Grades 9-12.

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