1. **State the problem:** Simplify or analyze the quadratic expression $5t^2 - 7t - 124$. 2. **Formula and rules:** This is a quadratic expression of the form $at^2 + bt + c$ where $a=5$, $b=-7$, and $c=-124$. 3. **Find the roots using the quadratic formula:** $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-7)^2 - 4 \times 5 \times (-124) = 49 + 2480 = 2529$$ 5. **Calculate the roots:** $$t = \frac{7 \pm \sqrt{2529}}{10}$$ 6. **Simplify the square root if possible:** $2529$ is not a perfect square, so roots remain as is. 7. **Final answer:** $$t = \frac{7 + \sqrt{2529}}{10} \quad \text{or} \quad t = \frac{7 - \sqrt{2529}}{10}$$ These are the solutions to the quadratic equation $5t^2 - 7t - 124 = 0$.