1. **State the problem:** Find the points of intersection between the parabola $$y = 4 - x^2$$ and the line $$y = -x + 2$$. 2. **Set the equations equal to find intersection points:** $$4 - x^2 = -x + 2$$ 3. **Rearrange the equation:** $$4 - x^2 = -x + 2 \implies 4 - x^2 + x - 2 = 0$$ $$-x^2 + x + 2 = 0$$ Multiply both sides by $$-1$$ to simplify: $$x^2 - x - 2 = 0$$ 4. **Factor the quadratic:** $$x^2 - x - 2 = (x - 2)(x + 1) = 0$$ 5. **Solve for $$x$$:** $$x - 2 = 0 \implies x = 2$$ $$x + 1 = 0 \implies x = -1$$ 6. **Find corresponding $$y$$ values by substituting into the line equation $$y = -x + 2$$:** - For $$x = 2$$: $$y = -2 + 2 = 0$$ - For $$x = -1$$: $$y = -(-1) + 2 = 1 + 2 = 3$$ 7. **Intersection points are:** $$(2, 0)$$ and $$(-1, 3)$$. **Final answer:** The parabola and the line intersect at points $$(-1, 3)$$ and $$(2, 0)$$.