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^6 + 18x^3 + 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 a rectangular prism, we use the formula:

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

where:
- [tex]$ l $[/tex] is the length,
- [tex]$ w $[/tex] is the width,
- [tex]$ h $[/tex] is the height.

For this problem:
- The length ([tex]$ l $[/tex]) is [tex]$ 4x $[/tex],
- The width ([tex]$ w $[/tex]) is [tex]$ 2x $[/tex],
- The height ([tex]$ h $[/tex]) is [tex]$ x^3 + 3x + 6 $[/tex].

To find the volume, we multiply these expressions together:

[tex]\[ V = (4x) \cdot (2x) \cdot (x^3 + 3x + 6) \][/tex]

First, multiply the length and width:

[tex]\[ 4x \times 2x = 8x^2 \][/tex]

Now, multiply this result by the height:

[tex]\[ V = 8x^2 \cdot (x^3 + 3x + 6) \][/tex]

Distribute [tex]$ 8x^2 $[/tex] to each term inside the parentheses:

1. [tex]$ 8x^2 \cdot x^3 = 8x^{5} $[/tex]
2. [tex]$ 8x^2 \cdot 3x = 24x^{3} $[/tex]
3. [tex]$ 8x^2 \cdot 6 = 48x^{2} $[/tex]

Combine all the terms to get the volume:

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

Therefore, the volume of the rectangular prism is:

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