Find the volume of a rectangular prism if the length is [tex]4x[/tex], the width is [tex]2x[/tex], and the height is [tex]x^3 + 3x + 6[/tex]. Use the formula [tex]V = l \cdot w \cdot h[/tex], where [tex]l[/tex] is length, [tex]w[/tex] is width, and [tex]h[/tex] is height, to find the volume.

A. [tex]6x^5 + 18x^3 + 36x^2[/tex]
B. [tex]6x^4 + 18x^2 + 36x^2[/tex]
C. [tex]8x^5 + 24x^3 + 48x^2[/tex]
D. [tex]8x^6 + 24x^3 + 48x^2[/tex]

To find the volume of the rectangular prism, we will use the formula for volume, [tex]$ V $[/tex], given by:

[tex]\[ V = l \cdot w \cdot h \][/tex]

where [tex]$ l $[/tex] is the length, [tex]$ w $[/tex] is the width, and [tex]$ h $[/tex] is the height of the prism.

Given:
- [tex]$ l = 4x $[/tex]
- [tex]$ w = 2x $[/tex]
- [tex]$ h = x^3 + 3x + 6 $[/tex]

Let's perform the multiplication step-by-step:

1. Multiply the length and the width:
[tex]\[
l \cdot w = (4x) \cdot (2x) = 8x^2
\][/tex]

2. Multiply the result with the height:
[tex]\[
V = (8x^2) \cdot (x^3 + 3x + 6)
\][/tex]

3. Distribute [tex]$ 8x^2 $[/tex] to each term in the height [tex]$ x^3 + 3x + 6 $[/tex]:
[tex]\[
V = 8x^2 \cdot x^3 + 8x^2 \cdot 3x + 8x^2 \cdot 6
\][/tex]

4. Perform each multiplication:
[tex]\[
8x^2 \cdot x^3 = 8x^{2+3} = 8x^5
\][/tex]
[tex]\[
8x^2 \cdot 3x = 24x^{2+1} = 24x^3
\][/tex]
[tex]\[
8x^2 \cdot 6 = 48x^2
\][/tex]

5. Combine all the terms to get the final volume:
[tex]\[
V = 8x^5 + 24x^3 + 48x^2
\][/tex]

So, the volume of the rectangular prism is [tex]$ 8x^5 + 24x^3 + 48x^2 $[/tex].

Among the options given, the correct one is:

[tex]\[ 8x^5 + 24x^3 + 48x^2 \][/tex]