1. **State the problem:** Solve the quadratic equation $$4x^2 + 16x + 7 = 0$$ using the quadratic formula. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation. 3. **Identify coefficients:** Here, $a = 4$, $b = 16$, and $c = 7$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 16^2 - 4 \times 4 \times 7 = 256 - 112 = 144$$ 5. **Compute the square root of the discriminant:** $$\sqrt{\Delta} = \sqrt{144} = 12$$ 6. **Apply the quadratic formula:** $$x = \frac{-16 \pm 12}{2 \times 4} = \frac{-16 \pm 12}{8}$$ 7. **Find the two solutions:** - For the plus sign: $$x_1 = \frac{-16 + 12}{8} = \frac{\cancel{-16} + 12}{8} = \frac{-4}{8} = -\frac{1}{2}$$ - For the minus sign: $$x_2 = \frac{-16 - 12}{8} = \frac{-28}{8} = -\frac{7}{2}$$ **Final answer:** The two solutions are $$x = -\frac{1}{2}$$ and $$x = -\frac{7}{2}$$.