1. **State the problem:** We are given a rectangle with an area of 12 cm² and a length of $2 + \sqrt{5}$ cm. We need to find the width, leaving the answer in surd form. 2. **Formula used:** The area $A$ of a rectangle is given by $$A = \text{length} \times \text{width}$$ 3. **Set up the equation:** Let the width be $w$. Then, $$12 = (2 + \sqrt{5}) \times w$$ 4. **Solve for $w$:** $$w = \frac{12}{2 + \sqrt{5}}$$ 5. **Rationalize the denominator:** Multiply numerator and denominator by the conjugate of the denominator $2 - \sqrt{5}$: $$w = \frac{12}{2 + \sqrt{5}} \times \frac{2 - \sqrt{5}}{2 - \sqrt{5}} = \frac{12(2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})}$$ 6. **Simplify the denominator using difference of squares:** $$(2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1$$ 7. **Substitute back:** $$w = \frac{12(2 - \sqrt{5})}{-1} = -12(2 - \sqrt{5})$$ 8. **Distribute the negative sign:** $$w = -24 + 12\sqrt{5}$$ 9. **Final answer:** The width in surd form is $$w = 12\sqrt{5} - 24$$