To find the volume of a rectangular prism, we use the formula:
[tex]\[ V = \text{length} \times \text{width} \times \text{height} \][/tex]
Given:
- Length ([tex]\( l \)[/tex]) = [tex]\( 4x \)[/tex]
- Width ([tex]\( w \)[/tex]) = [tex]\( 5x \)[/tex]
- Height ([tex]\( h \)[/tex]) = [tex]\( x^2 + x + 2 \)[/tex]
We substitute these values into the volume formula:
[tex]\[ V = (4x) \times (5x) \times (x^2 + x + 2) \][/tex]
First, multiply the length and width:
[tex]\[ 4x \times 5x = 20x^2 \][/tex]
Next, multiply this result by the height:
[tex]\[ V = 20x^2 \times (x^2 + x + 2) \][/tex]
Now, distribute [tex]\( 20x^2 \)[/tex] through each term in the polynomial [tex]\(x^2 + x + 2\)[/tex]:
[tex]\[ V = 20x^2 \cdot x^2 + 20x^2 \cdot x + 20x^2 \cdot 2 \][/tex]
This simplifies to:
[tex]\[ V = 20x^2 \cdot x^2 + 20x^2 \cdot x + 20x^2 \cdot 2 \][/tex]
[tex]\[ V = 20x^4 + 20x^3 + 40x^2 \][/tex]
Thus, the volume of the rectangular prism is:
[tex]\[ \boxed{20x^4 + 20x^3 + 40x^2} \][/tex]
However, notice that the choices given have different forms. Based on the available choices, the one that matches our final calculation is:
[tex]\[ 20x^2(x^2 + x + 2) \][/tex]
Rewriting our final result:
[tex]\[ 20x^4 + 20x^3 + 40x^2 \][/tex]
matches:
[tex]\[ 20x^2(x^2 + x + 2) \][/tex]
Given the choices, the correct answer expressed differently:
[tex]\[ 20x^2(x^2 + x + 2) \][/tex]
Thus the answer is:
[tex]\[ \boxed{20 x^2(x^2 + x + 2)}\][/tex]
