1. **State the problem:** Solve the quadratic equation $$m^2 + 24m + 63 = 0$$. 2. **Formula and rules:** We use the quadratic formula $$m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the equation is in the form $$ax^2 + bx + c = 0$$. 3. **Identify coefficients:** Here, $$a = 1$$, $$b = 24$$, and $$c = 63$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 24^2 - 4 \times 1 \times 63 = 576 - 252 = 324$$. 5. **Find the square root of the discriminant:** $$\sqrt{324} = 18$$. 6. **Apply the quadratic formula:** $$m = \frac{-24 \pm 18}{2 \times 1} = \frac{-24 \pm 18}{2}$$. 7. **Calculate the two solutions:** - For the plus sign: $$m = \frac{-24 + 18}{2} = \frac{-6}{2} = -3$$. - For the minus sign: $$m = \frac{-24 - 18}{2} = \frac{-42}{2} = -21$$. 8. **Final answer:** The solutions to the equation are $$m = -3$$ and $$m = -21$$.