1. **Problem statement:** Simplify the expression $(\sqrt{x} - 2\sqrt{y})(\sqrt{x} + 2\sqrt{y}) + (3\sqrt{x} + \sqrt{y})^2$ using distribution. 2. **Recall formulas:** - Difference of squares: $(a - b)(a + b) = a^2 - b^2$ - Square of a binomial: $(a + b)^2 = a^2 + 2ab + b^2$ 3. **Apply difference of squares:** $$ (\sqrt{x} - 2\sqrt{y})(\sqrt{x} + 2\sqrt{y}) = (\sqrt{x})^2 - (2\sqrt{y})^2 = x - 4y $$ 4. **Expand the square:** $$ (3\sqrt{x} + \sqrt{y})^2 = (3\sqrt{x})^2 + 2 \times 3\sqrt{x} \times \sqrt{y} + (\sqrt{y})^2 = 9x + 6\sqrt{xy} + y $$ 5. **Add the two results:** $$ x - 4y + 9x + 6\sqrt{xy} + y = (x + 9x) + (-4y + y) + 6\sqrt{xy} = 10x - 3y + 6\sqrt{xy} $$ 6. **Final answer:** $$ \boxed{10x - 3y + 6\sqrt{xy}} $$