1. **State the problem:** Rationalise the denominator of the expression $$\frac{4}{\sqrt{7} - \sqrt{11}}.$$\n\n2. **Recall the formula:** To rationalise a denominator with a difference of square roots, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $$\sqrt{7} - \sqrt{11}$$ is $$\sqrt{7} + \sqrt{11}$$.\n\n3. **Multiply numerator and denominator by the conjugate:**\n$$\frac{4}{\sqrt{7} - \sqrt{11}} \times \frac{\sqrt{7} + \sqrt{11}}{\sqrt{7} + \sqrt{11}} = \frac{4(\sqrt{7} + \sqrt{11})}{(\sqrt{7} - \sqrt{11})(\sqrt{7} + \sqrt{11})}.$$\n\n4. **Simplify the denominator using difference of squares:**\n$$ (\sqrt{7})^2 - (\sqrt{11})^2 = 7 - 11 = -4.$$\n\n5. **Rewrite the expression:**\n$$\frac{4(\sqrt{7} + \sqrt{11})}{-4}.$$\n\n6. **Cancel common factors:**\n$$\frac{\cancel{4}(\sqrt{7} + \sqrt{11})}{\cancel{-4}} = - (\sqrt{7} + \sqrt{11}).$$\n\n7. **Final answer:**\n$$-\sqrt{7} - \sqrt{11}.$$