1. **State the problem:** We need to find the number of tiles required to cover a rectangular bathroom wall. 2. **Given:** - Length of wall = $12 + 12\sqrt{3}$ meters - Width of wall = $1 + \sqrt{3}$ meters - Area of each tile = $4 + 2\sqrt{3}$ square meters 3. **Calculate the area of the wall:** $$\text{Area}_{wall} = (12 + 12\sqrt{3})(1 + \sqrt{3})$$ 4. **Expand the product:** $$= 12 \times 1 + 12 \times \sqrt{3} + 12\sqrt{3} \times 1 + 12\sqrt{3} \times \sqrt{3}$$ $$= 12 + 12\sqrt{3} + 12\sqrt{3} + 12 \times 3$$ $$= 12 + 24\sqrt{3} + 36$$ 5. **Simplify:** $$= (12 + 36) + 24\sqrt{3} = 48 + 24\sqrt{3}$$ 6. **Calculate the number of tiles needed:** $$\text{Number of tiles} = \frac{\text{Area}_{wall}}{\text{Area}_{tile}} = \frac{48 + 24\sqrt{3}}{4 + 2\sqrt{3}}$$ 7. **Simplify the fraction by rationalizing the denominator:** Multiply numerator and denominator by the conjugate of the denominator $4 - 2\sqrt{3}$: $$\frac{48 + 24\sqrt{3}}{4 + 2\sqrt{3}} \times \frac{4 - 2\sqrt{3}}{4 - 2\sqrt{3}} = \frac{(48 + 24\sqrt{3})(4 - 2\sqrt{3})}{(4 + 2\sqrt{3})(4 - 2\sqrt{3})}$$ 8. **Calculate denominator:** $$4^2 - (2\sqrt{3})^2 = 16 - 4 \times 3 = 16 - 12 = 4$$ 9. **Calculate numerator:** $$48 \times 4 - 48 \times 2\sqrt{3} + 24\sqrt{3} \times 4 - 24\sqrt{3} \times 2\sqrt{3}$$ $$= 192 - 96\sqrt{3} + 96\sqrt{3} - 24 \times 2 \times 3$$ $$= 192 + ( -96\sqrt{3} + 96\sqrt{3}) - 144$$ $$= 192 - 144 = 48$$ 10. **Divide numerator by denominator:** $$\frac{48}{4} = 12$$ **Final answer:** The school needs 12 tiles to cover the entire wall.