1. **State the problem:** Solve the quadratic equation $$t^2 - 4t - 40 = 0$$ for $t$. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-4$, and $c=-40$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 1 \times (-40) = 16 + 160 = 176$$ 4. **Apply the quadratic formula:** $$t = \frac{-(-4) \pm \sqrt{176}}{2 \times 1} = \frac{4 \pm \sqrt{176}}{2}$$ 5. **Simplify the square root:** $$\sqrt{176} = \sqrt{16 \times 11} = 4\sqrt{11}$$ 6. **Substitute back:** $$t = \frac{4 \pm 4\sqrt{11}}{2}$$ 7. **Simplify the fraction by canceling common factor 2:** $$t = \frac{\cancel{2} \times 2 \pm \cancel{2} \times 2\sqrt{11}}{\cancel{2} \times 1} = 2 \pm 2\sqrt{11}$$ 8. **Final solutions:** $$t = 2 + 2\sqrt{11} \quad \text{or} \quad t = 2 - 2\sqrt{11}$$