1. **State the problem:** Simplify the expression $$\frac{2x^2 - 4x + 1}{2x^4 - 6x^2 - 5x + 7}$$. 2. **Factor numerator and denominator if possible:** - Numerator: $2x^2 - 4x + 1$ - Denominator: $2x^4 - 6x^2 - 5x + 7$ 3. **Factor numerator:** Use the quadratic formula for $2x^2 - 4x + 1=0$: $$x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} = \frac{4 \pm \sqrt{16 - 8}}{4} = \frac{4 \pm \sqrt{8}}{4} = \frac{4 \pm 2\sqrt{2}}{4} = 1 \pm \frac{\sqrt{2}}{2}$$ So numerator factors as: $$2\left(x - \left(1 + \frac{\sqrt{2}}{2}\right)\right)\left(x - \left(1 - \frac{\sqrt{2}}{2}\right)\right)$$ 4. **Factor denominator:** Try factoring by grouping: $$2x^4 - 6x^2 - 5x + 7 = (2x^4 - 6x^2) + (-5x + 7) = 2x^2(x^2 - 3) -1(5x - 7)$$ No common factor emerges easily; denominator does not factor nicely with simple methods. 5. **Conclusion:** The numerator factors as above, denominator remains as is. 6. **Final simplified form:** $$\frac{2\left(x - \left(1 + \frac{\sqrt{2}}{2}\right)\right)\left(x - \left(1 - \frac{\sqrt{2}}{2}\right)\right)}{2x^4 - 6x^2 - 5x + 7}$$ No common factors to cancel. **Answer:** The expression cannot be simplified further.