1. **State the problem:** Simplify the expression $$\left[(1 - a)^2 + (2 - b)^2\right] + (2a - b)(2a + b) + (1 + a)^2 + 4b - (a - b)(a + b)$$. 2. **Recall formulas and rules:** - Square of a binomial: $$(x \pm y)^2 = x^2 \pm 2xy + y^2$$ - Product of sum and difference: $$(x - y)(x + y) = x^2 - y^2$$ 3. **Expand each term:** - $$(1 - a)^2 = 1 - 2a + a^2$$ - $$(2 - b)^2 = 4 - 4b + b^2$$ - $$(2a - b)(2a + b) = (2a)^2 - b^2 = 4a^2 - b^2$$ - $$(1 + a)^2 = 1 + 2a + a^2$$ - $$(a - b)(a + b) = a^2 - b^2$$ 4. **Substitute expansions back into the expression:** $$\left[1 - 2a + a^2 + 4 - 4b + b^2\right] + 4a^2 - b^2 + 1 + 2a + a^2 + 4b - (a^2 - b^2)$$ 5. **Combine like terms inside the brackets:** $$1 + 4 = 5$$ $$-2a$$ $$a^2$$ $$-4b$$ $$b^2$$ So bracket becomes $$5 - 2a + a^2 - 4b + b^2$$ 6. **Rewrite entire expression:** $$5 - 2a + a^2 - 4b + b^2 + 4a^2 - b^2 + 1 + 2a + a^2 + 4b - (a^2 - b^2)$$ 7. **Simplify by combining like terms:** - Combine constants: $$5 + 1 = 6$$ - Combine $a^2$ terms: $$a^2 + 4a^2 + a^2 = 6a^2$$ - Combine $b^2$ terms: $$b^2 - b^2 - (a^2 - b^2) = b^2 - b^2 - a^2 + b^2 = -a^2 + b^2$$ - Combine $a$ terms: $$-2a + 2a = 0$$ - Combine $b$ terms: $$-4b + 4b = 0$$ 8. **Substitute combined terms:** $$6 + 6a^2 - a^2 + b^2$$ 9. **Simplify further:** $$6 + (6a^2 - a^2) + b^2 = 6 + 5a^2 + b^2$$ **Final answer:** $$\boxed{6 + 5a^2 + b^2}$$