To find the volume of a trapezoidal prism, we need to use the formula for the volume of a prism, [tex]\( V = A \cdot \text{height} \)[/tex], where [tex]\( A \)[/tex] is the area of the base and [tex]\(\text{height} \)[/tex] is the perpendicular distance between the two bases.
From the given expressions for calculating the base area [tex]\( A \)[/tex]:
[tex]\[
\begin{array}{l}
A=\frac{1}{2}((x+4)+(x+2)) x \\
A=\frac{1}{2}(2 x+6) x \\
A=(x+3) x \\
A=x^2+3 x
\end{array}
\][/tex]
We have derived that [tex]\( A = x^2 + 3x \)[/tex].
Next, we need to consider the height of the trapezoidal prism, which is given as [tex]\( 2x \)[/tex].
The volume [tex]\( V \)[/tex] can then be calculated as:
[tex]\[
V = A \cdot \text{height}
\][/tex]
Substituting the values we have:
[tex]\[
V = (x^2 + 3x) \cdot 2x
\][/tex]
Multiplying the terms inside the parentheses by [tex]\( 2x \)[/tex]:
[tex]\[
V = 2x \cdot (x^2 + 3x)
\][/tex]
This expands to:
[tex]\[
V = 2x \cdot x^2 + 2x \cdot 3x
\][/tex]
Simplifying each term, we get:
[tex]\[
V = 2x^3 + 6x^2
\][/tex]
Thus, the expression that represents the volume of the trapezoidal prism is:
[tex]\[
2x^3 + 6x^2
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{2x^3 + 6x^2}
\][/tex]
