To find an equivalent expression for \( 4x^3 + 8x^2 + 12x \) using the Distributive Property, we first look for a common factor in each term of the expression.
The given expression is:
\[ 4x^3 + 8x^2 + 12x \]
We can factor out the greatest common factor, which is \( 4x \):
\[ 4x(x^2 + 2x + 3) \]
Now we need to compare this expression to the options provided:
1. \( 4x(4x + 5x) \) β This simplifies to \( 4x(9x) \), which does not match.
2. \( 4x(x^2 + 2x + 3) \) β This is the expression we arrived at, and it is equivalent.
3. \( (2x^2 + 4)(2x^2 + 3) \) β This does not match our expression when expanded.
4. \( (2x^2 + 4)(2x - 3) \) β This also does not match when expanded.
Thus, the correct answer is:
\[ \boxed{4x(x^2 + 2x + 3)} \]