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 with given dimensions, we use the formula for volume:

[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.

For this problem, the dimensions are:
- Length [tex]\( l = 4x \)[/tex]
- Width [tex]\( w = 2x \)[/tex]
- Height [tex]\( h = x^3 + 3x + 6 \)[/tex]

Substituting these into the formula, we get:

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

First, multiply the length and width:

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

Next, multiply this result by the height:

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

To do this multiplication, apply the distributive property:

1. Multiply [tex]\( 8x^2 \)[/tex] by [tex]\( x^3 \)[/tex]:
[tex]\[ 8x^2 \cdot x^3 = 8x^{5} \][/tex]

2. Multiply [tex]\( 8x^2 \)[/tex] by [tex]\( 3x \)[/tex]:
[tex]\[ 8x^2 \cdot 3x = 24x^{3} \][/tex]

3. Multiply [tex]\( 8x^2 \)[/tex] by [tex]\( 6 \)[/tex]:
[tex]\[ 8x^2 \cdot 6 = 48x^{2} \][/tex]

Add all the terms together to find the expression for the volume:

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

Therefore, the volume of the rectangular prism is:

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

This matches the option: [tex]\( 8x^5 + 24x^3 + 48x^2 \)[/tex].