To simplify the expression [tex]\(\sqrt{8x^3 + 24x^2 + 18x}\)[/tex], let's follow these steps:
1. Factor the expression inside the square root:
[tex]\[
8x^3 + 24x^2 + 18x
\][/tex]
We start by factoring out the greatest common factor (GCF) which is [tex]\(2x\)[/tex]:
[tex]\[
8x^3 + 24x^2 + 18x = 2x(4x^2 + 12x + 9)
\][/tex]
2. Further factor the quadratic expression:
We focus on factoring the quadratic [tex]\(4x^2 + 12x + 9\)[/tex]:
[tex]\[
4x^2 + 12x + 9 = (2x + 3)^2
\][/tex]
Therefore:
[tex]\[
8x^3 + 24x^2 + 18x = 2x(2x + 3)^2
\][/tex]
3. Take the square root of both sides:
[tex]\[
\sqrt{8x^3 + 24x^2 + 18x} = \sqrt{2x(2x + 3)^2}
\][/tex]
4. Simplify the square root expression:
We know that the square root of a product equals the product of the square roots, hence:
[tex]\[
\sqrt{2x(2x + 3)^2} = \sqrt{2} \cdot \sqrt{x} \cdot \sqrt{(2x + 3)^2}
\][/tex]
Since [tex]\(\sqrt{(2x + 3)^2} = |2x + 3|\)[/tex], we get:
[tex]\[
\sqrt{2} \cdot \sqrt{x} \cdot |2x + 3|
\][/tex]
If [tex]\(2x + 3\)[/tex] is non-negative, then [tex]\(|2x + 3| = 2x + 3\)[/tex], and if [tex]\(2x + 3\)[/tex] is negative, [tex]\(|2x + 3| = -(2x + 3)\)[/tex]. Usually, in the context of real numbers, we use the positive root and include the [tex]\(\pm\)[/tex] sign to cover both cases.
Thus:
[tex]\[
\sqrt{2} \cdot \sqrt{x} \cdot |2x + 3| = \pm(2x + 3)\sqrt{2x}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\pm(2x + 3)\sqrt{2x}}
\][/tex]
Which corresponds to option A: [tex]\(\pm(2x + 3) \sqrt{2 x}\)[/tex].
