To find an expression equivalent to \(4x^3 + 8x^2 + 12x\) using the Distributive Property, we can factor out the common term from each part of the expression.
1. Identify the common factor:
- Each term \(4x^3\), \(8x^2\), and \(12x\) has a common factor of \(4x\).
2. Factor out \(4x\):
\[
4x^3 + 8x^2 + 12x = 4x(x^2 + 2x + 3)
\]
Now let's look at the answer choices:
1. \(4x(x^2 + 2x + 3)\) - This matches our factored expression.
2. \((2x^2 + 4)(2x^2 + 3)\) - This is not equivalent.
3. \(4x(4x + 5x)\) - This equals \(4x(9x) = 36x^2\), which is not equivalent.
4. \((2x^2 + 4)(2x - 3)\) - This is not equivalent.
The correct equivalent expression according to the Distributive Property is:
\[ 4x(x^2 + 2x + 3) \]