Find the area of the shaded region.
\[
f(x)=3 x+2 x^{2}-x^{3}, g(x)=0
\]
The area is (Type an integer or a simplified fraction.)

Step 1 ：The area of the shaded region between two functions, f(x) and g(x), from a to b is given by the integral from a to b of |f(x) - g(x)| dx. In this case, g(x) = 0, so we need to find the integral from a to b of |f(x)| dx.

Step 2 ：However, we don't have the limits of integration (a and b). We need to find the x-values where f(x) intersects g(x), which are the solutions to the equation f(x) = g(x). Since g(x) = 0, we need to find the roots of the equation f(x) = 0.

Step 3 ：The roots of the equation f(x) = 0 are -1, 0, and 3. These are the x-values where the function f(x) intersects the x-axis (g(x) = 0). So, the limits of integration are -1, 0, and 3.

Step 4 ：However, since we are looking for the area, which is always positive, we need to take the absolute value of the function f(x) between these limits. We can split the integral into two parts: from -1 to 0 and from 0 to 3, and add the absolute values of these two integrals to get the total area.

Step 5 ：The area of the first part from -1 to 0 is \(\frac{7}{12}\) and the area of the second part from 0 to 3 is \(\frac{45}{4}\).

Step 6 ：Adding these two areas, we get the total area as \(\frac{71}{6}\).

Step 7 ：Final Answer: The area of the shaded region is \(\boxed{\frac{71}{6}}\).