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 $y$:

$$6 + x - x^2 = 2 - 2x$$

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:

$$x^2 - 3x - 4 = 0$$