To solve this problem, let's break down the given information and figure out the area of the shaded region step-by-step.

Given Information:

1. $BCDF$ is a rectangle.
2. The area of $\triangle ABE$ is $2 \, \text{cm}^2$.
3. The area of $\triangle BEF$ is $3 \, \text{cm}^2$.

Objective:

We need to find the area of the shaded part.

Approach:

1. Calculate the Area of the Entire Rectangle $BCDF$:

   • Since we don’t have the dimensions of $BCDF$, let's focus on using the areas of triangles provided to determine the remaining areas.

2. Determine the Area of $\triangle ADF$:

   • Notice that if we have the areas of two triangles within the rectangle, we can attempt to find the remaining area by calculating the overall area that includes $\triangle ADF$.

3. Calculate the Area of the Shaded Region:

   • From the diagram, it appears that the shaded region might be part of the rectangle $BCDF$ minus the areas of $\triangle ABE$ and $\triangle BEF$.

Since the problem may involve additional geometric relationships not directly visible, let's assume the shaded region is the area we’re left with after subtracting triangles we know.

By calculating this setup correctly, we can deduce the shaded area based on provided data.

I’ll proceed with calculations and show more detail if necessary.