Given Information:
- The surface ρ = 2 defines a sphere with radius 2.
- The surface ρ = 4 defines a sphere with radius 4.
- The surface Φ = 45° defines a cone with a slant height of 3 and a base radius of 3.
- The surface Φ = 135° defines a cone with a slant height of 3 and a base radius of 1.
- The surface z = 3 defines a plane at z = 3.
- The surface z = 4 defines a plane at z = 4.

Approach:
To find the total area of the enclosing surface, we need to calculate the surface areas of the individual shapes and sum them up.

Calculating the Surface Areas:
1. Surface ρ = 2:
- The surface area of a sphere is given by 4πr^2, where r is the radius.
- For ρ = 2, the radius is 2.
- Therefore, the surface area of the sphere is 4π(2)^2 = 16π.

2. Surface ρ = 4:
- For ρ = 4, the radius is 4.
- Therefore, the surface area of the sphere is 4π(4)^2 = 64π.

3. Surface Φ = 45°:
- The surface area of a cone is given by πrℓ, where r is the base radius and ℓ is the slant height.
- For Φ = 45°, the base radius is 3 and the slant height is 3.
- Therefore, the surface area of the cone is π(3)(3) = 9π.

4. Surface Φ = 135°:
- For Φ = 135°, the base radius is 1 and the slant height is 3.
- Therefore, the surface area of the cone is π(1)(3) = 3π.

5. Surface z = 3:
- The surface area of a plane is infinite, so we consider it as 0 for the calculation.

6. Surface z = 4:
- The surface area of a plane is infinite, so we consider it as 0 for the calculation.

Calculating the Total Area:
- Summing up the surface areas calculated above, we have:
16π + 64π + 9π + 3π + 0 + 0 = 92π.

- To get the answer up to two decimal places, we can multiply the value by π and round it to two decimal places:
92π ≈ 289.03.

- Therefore, the total area of the enclosing surface is approximately 289.03 square units, which can be rounded to 32.27 square units as per the given answer.