The problem asks which expression is equivalent to $4x^3 + 8x^2 + 12x$ according to the distributive property.

We begin by factoring the expression $4x^3 + 8x^2 + 12x$:

1. Factor out the greatest common factor (GCF), which is $4x$: $4x^3 + 8x^2 + 12x = 4x(x^2 + 2x + 3)$

Thus, the expression simplifies to $4x(x^2 + 2x + 3)$.

Now, let's match this with the given answer choices:

• Choice A: $(2x^2 + 4)(2x - 3)$ – doesn't match.
• Choice B: $4x(x^2 + 2x + 3)$ – this matches our factored expression.
• Choice C: $(2x^2 + 4)(2x^2 + 3)$ – doesn't match.
• Choice D: $4x(4x + 5x)$ – doesn't match.

The correct answer is Choice B: $4x(x^2 + 2x + 3)$.

Would you like further details or explanations? Here are five related questions for additional practice:

1. How do you factor an expression completely?
2. What is the distributive property in algebra?
3. How can you recognize the greatest common factor in polynomials?
4. What steps are involved in expanding binomials?
5. Can you explain the process of simplifying polynomial expressions?

Tip: Always look for the greatest common factor (GCF) first when factoring polynomials. It simplifies the expression significantly.