To solve this problem, let's first analyze the given expression and apply the Distributive Property to determine which of the provided options is equivalent to $4x^3 + 8x^2 + 12x$.

Step 1: Factor the Expression

Given expression: $4x^3 + 8x^2 + 12x$

We can factor out the greatest common factor (GCF) from all the terms. The GCF here is $4x$, so: $4x^3 + 8x^2 + 12x = 4x(x^2 + 2x + 3)$

Step 2: Compare with the Options

Now let's compare this factored form $4x(x^2 + 2x + 3)$ with the given options:

1. $4x(x^2 + 2x + 3)$
2. $(2x^2 + 4)(2x - 3)$
3. $(2x^2 + 4)(2x^2 + 3)$
4. $4x(4x + 5x)$

The first option, $4x(x^2 + 2x + 3)$, matches our factored form exactly.

Conclusion

The correct answer is the first option: $4x(x^2 + 2x + 3)$.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How would you apply the Distributive Property to factor $6x^3 + 9x^2 + 12x$?
2. What are the steps to factor a quadratic trinomial?
3. How can you verify that the factored form of an expression is correct?
4. What is the difference between factoring and expanding expressions?
5. How do you factor an expression when the GCF is a binomial?

Tip: When factoring, always start by finding the greatest common factor (