1. **State the problem:** Simplify the expression $$4[x - 4(y - 2[5y + 2])]$$. 2. **Recall the distributive property:** $$a(b + c) = ab + ac$$ and the rule for simplifying inside brackets step-by-step. 3. **Simplify inside the innermost brackets:** $$5y + 2$$ 4. **Multiply by 2:** $$2[5y + 2] = 2 \times 5y + 2 \times 2 = 10y + 4$$ 5. **Simplify inside the next bracket:** $$y - (10y + 4) = y - 10y - 4 = -9y - 4$$ 6. **Multiply by 4:** $$4(-9y - 4) = 4 \times -9y + 4 \times -4 = -36y - 16$$ 7. **Simplify inside the outer bracket:** $$x - 4(y - 2[5y + 2]) = x - (-36y - 16) = x + 36y + 16$$ 8. **Multiply by 4:** $$4[x - 4(y - 2[5y + 2])] = 4(x + 36y + 16) = 4x + 144y + 64$$ **Final answer:** $$4x + 144y + 64$$