1. **State the problem:** Solve the simultaneous equations: $$y = 2 - x$$ $$y = x^2 + 2x + 2$$ 2. **Set the equations equal:** Since both equal $y$, set them equal to each other: $$2 - x = x^2 + 2x + 2$$ 3. **Rearrange to form a quadratic equation:** Move all terms to one side: $$0 = x^2 + 2x + 2 - 2 + x$$ $$0 = x^2 + 3x$$ 4. **Simplify:** $$x^2 + 3x = 0$$ 5. **Factor the quadratic:** $$x(x + 3) = 0$$ 6. **Solve for $x$:** $$x = 0 \quad \text{or} \quad x = -3$$ 7. **Find corresponding $y$ values:** - For $x=0$: $$y = 2 - 0 = 2$$ - For $x=-3$: $$y = 2 - (-3) = 2 + 3 = 5$$ 8. **Final solutions:** $$(x, y) = (0, 2) \quad \text{and} \quad (-3, 5)$$