1. **State the problem:** Solve the quadratic equation $$4x^2 + 16x + 15 = 0$$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions for $$x$$ are given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $$a = 4$$, $$b = 16$$, and $$c = 15$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 16^2 - 4 \times 4 \times 15 = 256 - 240 = 16$$ 5. **Evaluate the square root of the discriminant:** $$\sqrt{16} = 4$$ 6. **Apply the quadratic formula:** $$x = \frac{-16 \pm 4}{2 \times 4} = \frac{-16 \pm 4}{8}$$ 7. **Find the two solutions:** - For the plus sign: $$x = \frac{-16 + 4}{8} = \frac{-12}{8} = -\frac{3}{2}$$ - For the minus sign: $$x = \frac{-16 - 4}{8} = \frac{-20}{8} = -\frac{5}{2}$$ **Final answer:** The solutions to the equation are $$x = -\frac{3}{2}$$ and $$x = -\frac{5}{2}$$.