1. **State the problem:** Find the surface area of a cylinder with base radius $r=2$ m and height $h=4$ m. 2. **Formula for surface area of a cylinder:** $$\text{Surface Area} = 2\pi r^2 + 2\pi rh$$ This formula includes the areas of the two circular bases ($2\pi r^2$) and the rectangular side (lateral surface area, $2\pi rh$). 3. **Calculate the area of the two bases:** $$2\pi r^2 = 2\pi (2)^2 = 2\pi \times 4 = 8\pi$$ 4. **Calculate the lateral surface area:** $$2\pi rh = 2\pi \times 2 \times 4 = 16\pi$$ 5. **Add the two parts to get total surface area:** $$8\pi + 16\pi = 24\pi$$ 6. **Include units:** The surface area is $24\pi$ square meters. **Final answer:** $$\boxed{24\pi\ \text{m}^2}$$