1. The problem is to simplify the expression $$\frac{4}{\sqrt{5} + 1}$$ by rationalising the denominator. 2. To rationalise the denominator, multiply both numerator and denominator by the conjugate of the denominator, which is $$\sqrt{5} - 1$$. 3. Multiply numerator and denominator: $$\frac{4}{\sqrt{5} + 1} \times \frac{\sqrt{5} - 1}{\sqrt{5} - 1} = \frac{4(\sqrt{5} - 1)}{(\sqrt{5} + 1)(\sqrt{5} - 1)}$$ 4. Simplify the denominator using the difference of squares formula: $$(\sqrt{5})^2 - 1^2 = 5 - 1 = 4$$ 5. So the expression becomes: $$\frac{4(\sqrt{5} - 1)}{4}$$ 6. Cancel the 4 in numerator and denominator: $$\sqrt{5} - 1$$ 7. Therefore, the simplified expression is $$\sqrt{5} - 1$$.