Question

According to the Distributive Property, which expression is equivalent to 4x^3+8x^2+12x
?(1 point)
Responses

(2x2+4)(2x2+3)
left parenthesis 2 x squared plus 4 right parenthesis left parenthesis 2 x squared plus 3 right parenthesis

4x(x2+2x+3)
4 x left parenthesis x squared plus 2 x plus 3 right parenthesis

4x(4x+5x)
4 x left parenthesis 4 x plus 5 x right parenthesis

(2x2+4)(2x−3)
left parenthesis 2 x squared plus 4 right parenthesis left parenthesis 2 x minus 3 right parenthesis

Answer 1

To determine which expression is equivalent to \(4x^3 + 8x^2 + 12x\), we can factor it using the Distributive Property.

First, we can factor out the greatest common factor (GCF) from the expression:

1. Identify the GCF of the coefficients: The GCF of 4, 8, and 12 is 4.
2. The variable part: All terms have at least \(x\), so we can factor out \(x\) too.

Thus, the GCF of the entire expression is \(4x\).

Now we can factor \(4x\) out from each term:

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

Now we can match it with the given options:

1. \((2x^2 + 4)(2x^2 + 3)\) — Not equivalent.
2. \(4x(x^2 + 2x + 3)\) — This matches our factored expression.
3. \(4x(4x + 5x)\) — Not equivalent.
4. \((2x^2 + 4)(2x - 3)\) — Not equivalent.

The correct answer is:

**\(4x(x^2 + 2x + 3)\)**.