1. **State the problem:** Calculate the value of the expression $15.66 - \left[\frac{1}{4} + 3.15\right]^2$. 2. **Recall the order of operations:** First, solve inside the brackets, then square the result, and finally subtract from 15.66. 3. **Calculate inside the brackets:** $$\frac{1}{4} + 3.15 = 0.25 + 3.15 = 3.4$$ 4. **Square the result:** $$3.4^2 = 3.4 \times 3.4 = 11.56$$ 5. **Subtract from 15.66:** $$15.66 - 11.56 = 4.1$$ **Final answer:** $4.1$