Step 1: Set the two equations equal to each other to find the points of intersection. The given equations are: $$y = 6 + x - x^2$$ $$y = 2 - 2x$$ Equating the expressions for $y$: $$6 + x - x^2 = 2 - 2x$$ Step 2: Rearrange the equation into the standard quadratic form $ax^2 + bx + c = 0$. Add $x^2$ to both sides: $$6 + x = 2 - 2x + x^2$$ Subtract $x$ from both sides: $$6 = 2 - 3x + x^2$$ Subtract $6$ from both sides: $$0 = x^2 - 3x - 4$$ The simplified quadratic equation is: $$x^2 - 3x - 4 = 0$$ The quadratic equation is $\boxed{x^2 - 3x - 4 = 0}$.