1. **State the problem:** Solve the quadratic equation $$2x^2 - 3x - 15 = -4x$$ using the quadratic formula and express the answer in simplest form. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$2x^2 - 3x - 15 + 4x = 0$$ Simplify the linear terms: $$2x^2 + ( -3x + 4x ) - 15 = 0$$ $$2x^2 + x - 15 = 0$$ 3. **Identify coefficients:** The quadratic equation is in the form $$ax^2 + bx + c = 0$$ where: $$a = 2, \quad b = 1, \quad c = -15$$ 4. **Quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$b^2 - 4ac = 1^2 - 4 \times 2 \times (-15) = 1 + 120 = 121$$ 6. **Substitute values into the formula:** $$x = \frac{-1 \pm \sqrt{121}}{2 \times 2} = \frac{-1 \pm 11}{4}$$ 7. **Find the two solutions:** - For the plus sign: $$x = \frac{-1 + 11}{4} = \frac{10}{4} = \frac{\cancel{10}}{\cancel{4}} = \frac{5}{2}$$ - For the minus sign: $$x = \frac{-1 - 11}{4} = \frac{-12}{4} = \frac{\cancel{-12}}{\cancel{4}} = -3$$ 8. **Final answer:** $$x = \frac{5}{2} \quad \text{or} \quad x = -3$$