1. **Stating the problem:** Simplify the expression $$\frac{1}{a} \sqrt{\frac{ab^{2}}{4} + 3b} \left[ \frac{1}{\sqrt{4a}} - a \sqrt{ab^{2}} \right].$$ 2. **Recall important formulas and rules:** - The square root of a product: $$\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}.$$ - Simplify square roots by factoring perfect squares. - Distribute multiplication over addition/subtraction. 3. **Simplify inside the square root:** $$\sqrt{\frac{ab^{2}}{4} + 3b} = \sqrt{\frac{a b^{2}}{4} + 3b}.$$ 4. **Simplify the bracket:** - First term: $$\frac{1}{\sqrt{4a}} = \frac{1}{2\sqrt{a}}.$$ - Second term: $$a \sqrt{ab^{2}} = a \sqrt{a} \sqrt{b^{2}} = a \sqrt{a} \cdot b = a b \sqrt{a}.$$ So the bracket becomes: $$\frac{1}{2\sqrt{a}} - a b \sqrt{a}.$$ 5. **Rewrite the entire expression:** $$\frac{1}{a} \sqrt{\frac{a b^{2}}{4} + 3b} \left( \frac{1}{2\sqrt{a}} - a b \sqrt{a} \right).$$ 6. **Distribute the multiplication:** $$= \frac{1}{a} \sqrt{\frac{a b^{2}}{4} + 3b} \cdot \frac{1}{2\sqrt{a}} - \frac{1}{a} \sqrt{\frac{a b^{2}}{4} + 3b} \cdot a b \sqrt{a}.$$ 7. **Simplify each term:** - First term: $$\frac{1}{a} \cdot \frac{1}{2\sqrt{a}} = \frac{1}{2 a \sqrt{a}}.$$ So first term is: $$\frac{1}{2 a \sqrt{a}} \sqrt{\frac{a b^{2}}{4} + 3b}.$$ - Second term: $$\frac{1}{a} \cdot a b \sqrt{a} = b \sqrt{a}.$$ So second term is: $$b \sqrt{a} \sqrt{\frac{a b^{2}}{4} + 3b}.$$ 8. **Final expression:** $$\frac{1}{2 a \sqrt{a}} \sqrt{\frac{a b^{2}}{4} + 3b} - b \sqrt{a} \sqrt{\frac{a b^{2}}{4} + 3b}.$$ 9. **Factor out the common square root:** $$\sqrt{\frac{a b^{2}}{4} + 3b} \left( \frac{1}{2 a \sqrt{a}} - b \sqrt{a} \right).$$ This is the simplified form of the original expression. **Answer:** $$\boxed{\sqrt{\frac{a b^{2}}{4} + 3b} \left( \frac{1}{2 a \sqrt{a}} - b \sqrt{a} \right)}.$$