1. **State the problem:** Solve the equation $w(w+4) = -8$ using the quadratic formula and express the solution set in exact simplest form. 2. **Rewrite the equation:** Expand and bring all terms to one side: $$w^2 + 4w = -8$$ $$w^2 + 4w + 8 = 0$$ 3. **Identify coefficients:** For the quadratic equation $aw^2 + bw + c = 0$, here: $$a = 1, \quad b = 4, \quad c = 8$$ 4. **Quadratic formula:** $$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate discriminant:** $$b^2 - 4ac = 4^2 - 4 \times 1 \times 8 = 16 - 32 = -16$$ 6. **Since the discriminant is negative, solutions are complex:** $$w = \frac{-4 \pm \sqrt{-16}}{2}$$ 7. **Simplify the square root of negative number:** $$\sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4i$$ 8. **Substitute back:** $$w = \frac{-4 \pm 4i}{2}$$ 9. **Simplify the fraction:** $$w = \frac{\cancel{-4} \pm \cancel{4}i}{\cancel{2} \times 1} = -2 \pm 2i$$ 10. **Final solution set:** $$\boxed{\{ -2 + 2i, -2 - 2i \}}$$