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]
