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]

We are given the dimensions of the rectangular prism as follows:

- Length: [tex]$$4x$$[/tex]
- Width: [tex]$$2x$$[/tex]
- Height: [tex]$$x^3 + 3x + 6$$[/tex]

The formula for the volume [tex]$$V$$[/tex] of a rectangular prism is:

[tex]$$
V = \text{length} \times \text{width} \times \text{height}
$$[/tex]

Step 1: Multiply the Length and Width

First, we multiply the length and width:

[tex]$$
\text{Length} \times \text{Width} = (4x) \times (2x) = 8x^2
$$[/tex]

Step 2: Multiply by the Height

Next, we multiply the result from Step 1 by the height:

[tex]$$
V = 8x^2 \times \left( x^3 + 3x + 6 \right)
$$[/tex]

Step 3: Expand the Expression

Distribute [tex]$$8x^2$$[/tex] across the terms inside the parentheses:

- Multiply [tex]$$8x^2 \times x^3$$[/tex]:

[tex]$$
8x^2 \times x^3 = 8x^{2+3} = 8x^5
$$[/tex]

- Multiply [tex]$$8x^2 \times 3x$$[/tex]:

[tex]$$
8x^2 \times 3x = 24x^{2+1} = 24x^3
$$[/tex]

- Multiply [tex]$$8x^2 \times 6$$[/tex]:

[tex]$$
8x^2 \times 6 = 48x^2
$$[/tex]

Putting all these together, the volume is:

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

Thus, the volume of the rectangular prism is:

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