1. **State the problem:** Solve the quadratic equation $$w^2 + 12w - 40 = 0$$ and find which of the given options is a solution. 2. **Formula used:** To solve a quadratic equation $$aw^2 + bw + c = 0$$, use the quadratic formula: $$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=12$, and $c=-40$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 12^2 - 4 \times 1 \times (-40) = 144 + 160 = 304$$ 4. **Find the roots:** $$w = \frac{-12 \pm \sqrt{304}}{2}$$ Since $\sqrt{304} = \sqrt{16 \times 19} = 4\sqrt{19}$, we have: $$w = \frac{-12 \pm 4\sqrt{19}}{2} = -6 \pm 2\sqrt{19}$$ 5. **Solutions:** $$w_1 = -6 + 2\sqrt{19}$$ $$w_2 = -6 - 2\sqrt{19}$$ 6. **Check options:** - Option A: $6 - 2\sqrt{19}$ (not equal to either root) - Option B: $2\sqrt{19}$ (not equal) - Option C: $\sqrt{19}$ (not equal) - Option D: $-6 + 2\sqrt{19}$ (matches $w_1$) **Final answer:** Option D is a solution to the equation.