1. **State the problem:** Factorise the expression $$11y^2 + 11$$ completely. 2. **Identify the common factor:** Both terms have a common factor of 11. 3. **Use the distributive property:** Factor out 11 from each term: $$11y^2 + 11 = 11(y^2 + 1)$$ 4. **Check if further factorisation is possible:** The expression inside the parentheses is $$y^2 + 1$$, which is a sum of squares and cannot be factorised further over the real numbers. 5. **Final answer:** $$\boxed{11(y^2 + 1)}$$