1. **State the problem:** Simplify the expression $$23 - \frac{3}{4} (\sqrt{150a})^{2} \cdot 8a \sqrt{242a^{2}}$$. 2. **Recall important rules:** - The square of a square root simplifies as $$(\sqrt{x})^{2} = x$$. - Simplify radicals by factoring out perfect squares. - Multiply coefficients and variables separately. 3. **Simplify inside the expression:** $$(\sqrt{150a})^{2} = 150a$$ 4. **Rewrite the expression:** $$23 - \frac{3}{4} \times 150a \times 8a \times \sqrt{242a^{2}}$$ 5. **Simplify the radical:** $$\sqrt{242a^{2}} = \sqrt{121 \times 2 \times a^{2}} = 11a \sqrt{2}$$ 6. **Substitute back:** $$23 - \frac{3}{4} \times 150a \times 8a \times 11a \sqrt{2}$$ 7. **Multiply constants and variables:** $$\frac{3}{4} \times 150 \times 8 \times 11 = \frac{3}{4} \times 13200 = \frac{3 \times 13200}{4}$$ 8. **Simplify the fraction:** $$\frac{3 \times 13200}{4} = \frac{39600}{4} = 9900$$ 9. **Multiply variables:** $$a \times a \times a = a^{3}$$ 10. **Combine all:** $$23 - 9900 a^{3} \sqrt{2}$$ **Final answer:** $$\boxed{23 - 9900 a^{3} \sqrt{2}}$$