1. **State the problem:** Simplify the expression $$\frac{18x^2}{x - 1} \div \frac{x}{2x - 2}$$. 2. **Rewrite the division as multiplication by the reciprocal:** $$\frac{18x^2}{x - 1} \times \frac{2x - 2}{x}$$ 3. **Factor where possible:** Note that $$2x - 2 = 2(x - 1)$$, so the expression becomes: $$\frac{18x^2}{x - 1} \times \frac{2(x - 1)}{x}$$ 4. **Multiply the numerators and denominators:** $$\frac{18x^2 \times 2(x - 1)}{(x - 1) \times x}$$ 5. **Simplify by canceling common factors:** Cancel $$x - 1$$ in numerator and denominator: $$\frac{18x^2 \times \cancel{2(x - 1)}}{\cancel{(x - 1)} \times x} = \frac{18x^2 \times 2}{x}$$ 6. **Multiply numerator:** $$\frac{36x^2}{x}$$ 7. **Simplify the fraction by canceling $$x$$:** $$\frac{36\cancel{x^2}}{\cancel{x}} = 36x$$ **Final answer:** $$36x$$