To find the expression that represents the volume of a right pyramid with a square base where the base length is [tex]\(x\)[/tex] inches and the height is two inches longer than the base length, we can follow these steps:
1. Identify given quantities:
- Base length of the pyramid = [tex]\(x\)[/tex] inches.
- Height of the pyramid = [tex]\(x + 2\)[/tex] inches (since it's two inches longer than the length of the base).
2. Formula for the volume of a pyramid:
The formula to calculate the volume of a pyramid is given by:
[tex]\[
\text{Volume} = \frac{\text{Base Area} \times \text{Height}}{3}
\][/tex]
3. Calculate the base area:
For a pyramid with a square base, the area of the base ([tex]\(A\)[/tex]) is the side length squared:
[tex]\[
\text{Base Area} = x^2
\][/tex]
4. Substitute the known values into the volume formula:
Now, substitute the base area and the height into the volume formula:
[tex]\[
\text{Volume} = \frac{x^2 \times (x + 2)}{3}
\][/tex]
Simplifying this expression, we get:
[tex]\[
\text{Volume} = \frac{x^2(x + 2)}{3}
\][/tex]
So, the expression that represents the volume of the pyramid in terms of [tex]\(x\)[/tex] is:
[tex]\[
\boxed{\frac{x^2(x + 2)}{3}}
\][/tex]
This corresponds to the first option given in the list:
[tex]\[
\frac{x^2(x+2)}{3} \text{ cubic inches}
\][/tex]
Thus, the correct expression representing the volume of the pyramid is:
[tex]\[
\frac{x^2(x + 2)}{3} \text{ cubic inches}
\][/tex]
