1. **Stating the problem:** Simplify the expression $$\frac{3\sqrt{x} - 5}{2} \times 6\sqrt{x} - 5$$. 2. **Rewrite the expression clearly:** The expression is $$\left(\frac{3\sqrt{x} - 5}{2}\right) \times (6\sqrt{x} - 5)$$. 3. **Use the distributive property:** Multiply the numerators and then divide by the denominator. 4. **Multiply the numerators:** $$ (3\sqrt{x} - 5)(6\sqrt{x} - 5) = 3\sqrt{x} \times 6\sqrt{x} - 3\sqrt{x} \times 5 - 5 \times 6\sqrt{x} + 5 \times 5 $$ 5. **Calculate each term:** - $3\sqrt{x} \times 6\sqrt{x} = 18x$ - $3\sqrt{x} \times 5 = 15\sqrt{x}$ - $5 \times 6\sqrt{x} = 30\sqrt{x}$ - $5 \times 5 = 25$ 6. **Combine terms:** $$18x - 15\sqrt{x} - 30\sqrt{x} + 25 = 18x - 45\sqrt{x} + 25$$ 7. **Put it all over the denominator 2:** $$\frac{18x - 45\sqrt{x} + 25}{2}$$ 8. **Final simplified expression:** $$\boxed{\frac{18x - 45\sqrt{x} + 25}{2}}$$ This is the simplified form of the given expression.