1. **State the problem:** Solve the quadratic expression $m^2 - 10m - 11$ for its roots. 2. **Recall the quadratic formula:** For any quadratic 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. 3. **Identify coefficients:** Here, $a = 1$, $b = -10$, and $c = -11$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-10)^2 - 4 \times 1 \times (-11) = 100 + 44 = 144$$ 5. **Apply the quadratic formula:** $$m = \frac{-(-10) \pm \sqrt{144}}{2 \times 1} = \frac{10 \pm 12}{2}$$ 6. **Find the two roots:** - For the plus sign: $$m = \frac{10 + 12}{2} = \frac{22}{2} = 11$$ - For the minus sign: $$m = \frac{10 - 12}{2} = \frac{\cancel{10 - 12}}{2} = \frac{-2}{2} = -1$$ 7. **Final answer:** The roots of the quadratic $m^2 - 10m - 11$ are $$m = 11 \text{ and } m = -1$$