1. **State the problem:** Simplify the expression $$\frac{4x^2 - 2x + 8x - 4}{u x^2 + 1 - u x}$$ and solve the equation $$\frac{2x^2 - 2x - 4}{u x^2 + 1 - u x} = 1$$ for $x$. 2. **Simplify the numerator of the first fraction:** Combine like terms in the numerator: $$4x^2 - 2x + 8x - 4 = 4x^2 + 6x - 4$$ 3. **Rewrite the denominator:** The denominator is: $$u x^2 + 1 - u x$$ 4. **Simplify the first fraction:** The simplified fraction is: $$\frac{4x^2 + 6x - 4}{u x^2 + 1 - u x}$$ 5. **Solve the equation:** $$\frac{2x^2 - 2x - 4}{u x^2 + 1 - u x} = 1$$ Multiply both sides by the denominator to clear the fraction: $$2x^2 - 2x - 4 = 1 \times (u x^2 + 1 - u x)$$ 6. **Expand the right side:** $$2x^2 - 2x - 4 = u x^2 + 1 - u x$$ 7. **Bring all terms to one side:** $$2x^2 - 2x - 4 - u x^2 - 1 + u x = 0$$ Simplify: $$ (2 - u) x^2 + (-2 + u) x - 5 = 0$$ 8. **Solve the quadratic equation:** The quadratic is: $$ (2 - u) x^2 + (-2 + u) x - 5 = 0$$ Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = 2 - u$, $b = -2 + u$, and $c = -5$. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-2 + u)^2 - 4 (2 - u)(-5)$$ $$= (u - 2)^2 + 20 (2 - u)$$ Simplify: $$= (u - 2)^2 + 40 - 20u = u^2 - 4u + 4 + 40 - 20u = u^2 - 24u + 44$$ 9. **Final solution for $x$:** $$x = \frac{-( -2 + u) \pm \sqrt{u^2 - 24u + 44}}{2 (2 - u)} = \frac{2 - u \pm \sqrt{u^2 - 24u + 44}}{2 (2 - u)}$$ 10. **Note:** The solution depends on the parameter $u$ and the discriminant must be non-negative for real solutions. **Final answers:** - Simplified fraction: $$\frac{4x^2 + 6x - 4}{u x^2 + 1 - u x}$$ - Solutions to the equation: $$x = \frac{2 - u \pm \sqrt{u^2 - 24u + 44}}{2 (2 - u)}$$