To determine which expression is equivalent to \( 4x^3 + 8x^2 + 12x \) using the Distributive Property, we can factor out the common term in the original expression.
The greatest common factor of the terms \( 4x^3 \), \( 8x^2 \), and \( 12x \) is \( 4x \).
Let's factor it out:
\[ 4x^3 + 8x^2 + 12x = 4x(x^2 + 2x + 3) \]
Now we can analyze the given options:
- (2x^2 + 4)(2x^2 + 3) - This does not match because there is no \( 4x \) common factor.
- 4x(x^2 + 2x + 3) - This matches with our factored expression.
- (2x^2 + 4)(2x - 3) - This does not match because it introduces different factors.
- 4x(4x + 5x) - This simplifies to \( 4x(9x) \), which does not match.
The correct response is: 4x(x^2 + 2x + 3).