1. **State the problem:** Solve the equation $2x(x-4) = (1-x)(x+2)$. 2. **Write down the equation:** $$2x(x-4) = (1-x)(x+2)$$ 3. **Expand both sides:** Left side: $$2x^2 - 8x$$ Right side: $$1 \cdot x + 1 \cdot 2 - x \cdot x - x \cdot 2 = x + 2 - x^2 - 2x = -x^2 - x + 2$$ 4. **Rewrite the equation:** $$2x^2 - 8x = -x^2 - x + 2$$ 5. **Bring all terms to one side:** $$2x^2 - 8x + x^2 + x - 2 = 0$$ Simplify: $$3x^2 - 7x - 2 = 0$$ 6. **Use the quadratic formula:** For $ax^2 + bx + c = 0$, $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=3$, $b=-7$, $c=-2$. 7. **Calculate the discriminant:** $$\Delta = (-7)^2 - 4 \cdot 3 \cdot (-2) = 49 + 24 = 73$$ 8. **Find the roots:** $$x = \frac{7 \pm \sqrt{73}}{6}$$ 9. **Final answer:** $$x_1 = \frac{7 + \sqrt{73}}{6}, \quad x_2 = \frac{7 - \sqrt{73}}{6}$$