This problem involves calculating the surface area (SA) of a composite 3D shape formed by combining two rectangular prisms. Here's how we approach the problem:

Steps:

1. Understand the shape:

   • The blue prism has dimensions $12 \, \text{cm} \times 8 \, \text{cm} \times 2 \, \text{cm}$.
   • The red prism has dimensions $6 \, \text{cm} \times 8 \, \text{cm} \times 3 \, \text{cm}$.
   • These two shapes are combined, and the overlapping parts are not double-counted in the surface area.

2. Surface Area of Individual Shapes: The surface area of a rectangular prism is given by: $\text{SA of a prism} = 2lw + 2lh + 2wh$ For the blue prism: $\text{SA (blue)} = 2(12 \cdot 8) + 2(12 \cdot 2) + 2(8 \cdot 2) = 192 + 48 + 32 = 272 \, \text{cm}^2$ For the red prism: $\text{SA (red)} = 2(6 \cdot 8) + 2(6 \cdot 3) + 2(8 \cdot 3) = 96 + 36 + 48 = 180 \, \text{cm}^2$

3. Removing Overlapping Areas: Since the two prisms are combined, the overlapping part (common face) needs to be excluded. The common face is a rectangle of dimensions $8 \, \text{cm} \times 8 \, \text{cm}$: $\text{Area of overlap} = 8 \cdot 8 = 64 \, \text{cm}^2$ This area is counted twice (once for each prism), so we subtract it once.

4. Total Surface Area: Combine the surface areas and subtract the overlap: $\text{Total SA} = \text{SA (blue)} + \text{SA (red)} - \text{Area of overlap}$ Substituting values: $\text{Total SA} = 272 + 180 - 64 = 388 \, \text{cm}^2$

Final Answer:

The total surface area of the composite shape is: $\boxed{388 \, \text{cm}^2}$

Would you like additional clarifications or a step-by-step breakdown?

Here are some follow-up questions to deepen your understanding:

1. How is the formula for surface area derived for a rectangular prism?
2. What would change if the overlapping area were larger or smaller?
3. Can you find the volume of this composite shape?
4. How would the process differ if the prisms were not aligned on one side?
5. Why is the overlapping area subtracted only once?

Tip: When solving composite surface area problems, always visualize the overlapping sections to avoid double-counting.