1. **State the problem:** Simplify the expression $$\frac{\frac{(x - 2)^2}{2(x^2 - 5x + 4)}}{\frac{x^2 - 4}{4x - 10}}$$. 2. **Rewrite the complex fraction:** Dividing by a fraction is the same as multiplying by its reciprocal, so $$\frac{\frac{(x - 2)^2}{2(x^2 - 5x + 4)}}{\frac{x^2 - 4}{4x - 10}} = \frac{(x - 2)^2}{2(x^2 - 5x + 4)} \times \frac{4x - 10}{x^2 - 4}.$$ 3. **Factor all polynomials:** - Factor $x^2 - 5x + 4$ as $(x - 4)(x - 1)$. - Factor $x^2 - 4$ as $(x - 2)(x + 2)$. - Factor $4x - 10$ as $2(2x - 5)$. So the expression becomes $$\frac{(x - 2)^2}{2(x - 4)(x - 1)} \times \frac{2(2x - 5)}{(x - 2)(x + 2)}.$$ 4. **Multiply the numerators and denominators:** $$\frac{(x - 2)^2 \times 2(2x - 5)}{2(x - 4)(x - 1) \times (x - 2)(x + 2)}.$$ 5. **Cancel common factors:** - The factor 2 appears in numerator and denominator, so cancel: $$\frac{(x - 2)^2 \times \cancel{2}(2x - 5)}{\cancel{2}(x - 4)(x - 1)(x - 2)(x + 2)} = \frac{(x - 2)^2 (2x - 5)}{(x - 4)(x - 1)(x - 2)(x + 2)}.$$ - Cancel one $(x - 2)$ from numerator and denominator: $$\frac{\cancel{(x - 2)} (x - 2) (2x - 5)}{(x - 4)(x - 1) \cancel{(x - 2)} (x + 2)} = \frac{(x - 2)(2x - 5)}{(x - 4)(x - 1)(x + 2)}.$$ 6. **Final simplified expression:** $$\boxed{\frac{(x - 2)(2x - 5)}{(x - 4)(x - 1)(x + 2)}}.$$ This is the simplest form of the given expression after factoring and canceling common terms.