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To identify which expression is equivalent to \( 4x^3 + 8x^2 + 12x \) using the Distributive Property, we can first factor out the common factor \( 4x \) from the given expression.
Starting with \( 4x^3 + 8x^2 + 12x \):
1. Factor out \( 4x \):
\[
4x(x^2 + 2x + 3)
\]
Now we need to compare this expression with the provided options:
- \( (2x^2 + 4)(2x - 3) \)
- \( 4x(4x + 5x) \)
- \( (2x^2 + 4)(2x^2 + 3) \)
- \( 4x(x^2 + 2x + 3) \)
The only expression that matches our factored form of \( 4x(x^2 + 2x + 3) \) is:
**\( 4x(x^2 + 2x + 3) \)**
Thus, the correct answer is:
**\( 4x(x^2 + 2x + 3) \)**.