Question

According to the Distributive Property which expression is equivalent to
$4x^{3}+8x^{2}+12x$ ? (1 point)
$(2x^{2}+4)(2x^{2}+3)$
$4x(4x+5x)$
$4x(x^{2}+2x+3)$
$(2x^{2}+4)(2x-3)$

Answer 1

To simplify the expression \( 4x^{3} + 8x^{2} + 12x \) using the Distributive Property, we can factor out the greatest common factor (GCF).

First, we notice that each term in the expression \( 4x^{3} + 8x^{2} + 12x \) has a common factor of \( 4x \):

\[
4x^{3} = 4x \cdot x^{2}
\]
\[
8x^{2} = 4x \cdot 2x
\]
\[
12x = 4x \cdot 3
\]

So, we can factor out \( 4x \):

\[
4x^{3} + 8x^{2} + 12x = 4x(x^{2} + 2x + 3)
\]

Now, we look for an equivalent expression:

- \( (2x^{2}+4)(2x^{2}+3) \) is not equivalent.
- \( 4x(4x+5x) \) simplifies to \( 4x(9x) = 36x^2 \), which is not equivalent.
- \( 4x(x^{2}+2x+3) \) is what we obtained.
- \( (2x^{2}+4)(2x-3) \) is also not equivalent.

The correct equivalence is:

\[
\boxed{4x(x^{2}+2x+3)}
\]