What is 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 $[/tex] is the width, and [tex]$ h $[/tex] is the height.

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

Let's compute the volume step-by-step:

1. Multiply the Length and Width:
[tex]\[
l \cdot w = 4x \cdot 2x = 8x^2
\][/tex]

2. Multiply by the Height to find the Volume:
[tex]\[
V = 8x^2 \cdot (x^3 + 3x + 6)
\][/tex]

3. Distribute [tex]$ 8x^2 $[/tex] Across Each Term in the Parentheses:

- First Term:
[tex]\[
8x^2 \cdot x^3 = 8x^{2+3} = 8x^5
\][/tex]

- Second Term:
[tex]\[
8x^2 \cdot 3x = 24x^{2+1} = 24x^3
\][/tex]

- Third Term:
[tex]\[
8x^2 \cdot 6 = 48x^2
\][/tex]

4. Combine These Results:
[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].

Therefore, the correct answer is:

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