To solve the problem of finding the perimeter of the smallest rectangle that can contain both given rectangles, we follow these steps:

Step 1: Calculate the total area of the two rectangles

The first rectangle (12x8) has area: (12 \times 8 = 96)
The second rectangle (9x12) has area: (9 \times 12 = 108)
Total area: (96 + 108 = 204)

Step 2: Find the optimal arrangement

We need an enclosing rectangle with area ≥204 and minimal perimeter.

Key observation: Rotate the rectangles to align their common sides.

• Rotate the second rectangle to 12x9 (swap length and width).
• Stack the first rectangle (12x8) and rotated second rectangle (12x9) vertically:
  • Width of enclosing rectangle = max(12,12) = 12
  • Height = 8 + 9 = 17
  • Area = (12 \times 17 = 204) (exactly the total area, no wasted space).

Step 3: Calculate the perimeter

Perimeter = (2 \times (width + height) = 2 \times (12 + 17) = 58)

Answer: (\boxed{58})

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