1. **State the problem:** Simplify the expression $$- \frac{1}{4} \times \left( \frac{5}{8} + \frac{7}{8} \right) \div 2 \frac{2}{5} + \frac{5}{6}$$ using BIDMAS (Brackets, Indices, Division and Multiplication, Addition and Subtraction). 2. **Apply brackets first:** Calculate inside the parentheses: $$\frac{5}{8} + \frac{7}{8} = \frac{5+7}{8} = \frac{12}{8} = \frac{3}{2}$$ 3. **Rewrite the expression:** $$- \frac{1}{4} \times \frac{3}{2} \div 2 \frac{2}{5} + \frac{5}{6}$$ 4. **Convert mixed number to improper fraction:** $$2 \frac{2}{5} = \frac{2 \times 5 + 2}{5} = \frac{12}{5}$$ 5. **Rewrite expression with improper fraction:** $$- \frac{1}{4} \times \frac{3}{2} \div \frac{12}{5} + \frac{5}{6}$$ 6. **Perform multiplication and division from left to right:** First multiply: $$- \frac{1}{4} \times \frac{3}{2} = - \frac{3}{8}$$ Then divide by $$\frac{12}{5}$$ which is equivalent to multiplying by its reciprocal: $$- \frac{3}{8} \times \frac{5}{12} = - \frac{15}{96} = - \frac{5}{32}$$ 7. **Add the last fraction:** $$- \frac{5}{32} + \frac{5}{6}$$ Find common denominator: $$\text{LCM of } 32 \text{ and } 6 = 96$$ Convert fractions: $$- \frac{5}{32} = - \frac{15}{96}$$ $$\frac{5}{6} = \frac{80}{96}$$ Add: $$- \frac{15}{96} + \frac{80}{96} = \frac{65}{96}$$ 8. **Final answer:** $$\boxed{\frac{65}{96}}$$