1. **State the problem:** Simplify the expression $$(\sqrt{2} - 1)^2 - (\sqrt{2} + 3)(\sqrt{2} + 2).$$ 2. **Recall formulas and rules:** - Square of a binomial: $$(a - b)^2 = a^2 - 2ab + b^2$$ - Product of binomials: $$(a + b)(c + d) = ac + ad + bc + bd$$ 3. **Expand the first term:** $$ (\sqrt{2} - 1)^2 = (\sqrt{2})^2 - 2 \times \sqrt{2} \times 1 + 1^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} $$ 4. **Expand the second term:** $$ (\sqrt{2} + 3)(\sqrt{2} + 2) = (\sqrt{2})(\sqrt{2}) + (\sqrt{2})(2) + 3(\sqrt{2}) + 3(2) = 2 + 2\sqrt{2} + 3\sqrt{2} + 6 $$ 5. **Combine like terms in the second term:** $$ 2 + (2\sqrt{2} + 3\sqrt{2}) + 6 = 2 + 5\sqrt{2} + 6 = 8 + 5\sqrt{2} $$ 6. **Substitute back into the original expression:** $$ (3 - 2\sqrt{2}) - (8 + 5\sqrt{2}) $$ 7. **Distribute the minus sign:** $$ 3 - 2\sqrt{2} - 8 - 5\sqrt{2} = (3 - 8) + (-2\sqrt{2} - 5\sqrt{2}) = -5 - 7\sqrt{2} $$ **Final answer:** $$ \boxed{-5 - 7\sqrt{2}} $$