Answer
The volume of the provided figure as a polynomial function in standard form is:
\[V\left( x \right)={{x}^{3}}+7{{x}^{2}}-3x\].
Work Step by Step
Consider the provided figure:
There are two rectangles -- one is the outer rectangle and the second is the inner rectangle which is removed from the outer rectangle.
The length, width and height of the outer rectangle is $l=x,w=x+3$ and $h=2x-1$.
The length, width and height of the outer rectangle is $l=x,w=x$ and $h=\left( 2x-1 \right)-\left( x+1 \right)$.
Now, the volume of the provided figure is the difference of the volume of both the outer and inner rectangles.
$\begin{align}
& V\left( x \right)=\left[ \left( x \right)\left( 2x-1 \right)\left( x+3 \right) \right]-\left[ \left( x \right)\left( x \right)\left[ \left( 2x-1 \right)-\left( x+1 \right) \right] \right] \\
& =\left( x \right)\left( 2{{x}^{2}}+5x-3 \right)-{{x}^{2}}\left( x-2 \right) \\
& =2{{x}^{3}}+5{{x}^{2}}-3x-{{x}^{3}}+2{{x}^{2}} \\
& ={{x}^{3}}+7{{x}^{2}}-3x
\end{align}$
Hence, the volume of the provided figure as a polynomial function in standard form is $V\left( x \right)={{x}^{3}}+7{{x}^{2}}-3x$.
