1. **State the problem:** Find two numbers that multiply to 108 and add to -31. 2. **Formula and rules:** Let the two numbers be $x$ and $y$. Then: $$xy = 108$$ $$x + y = -31$$ 3. **Use the sum and product to form a quadratic equation:** $$t^2 - (x+y)t + xy = 0$$ Substitute the values: $$t^2 - (-31)t + 108 = 0$$ $$t^2 + 31t + 108 = 0$$ 4. **Solve the quadratic equation:** Calculate the discriminant: $$\Delta = b^2 - 4ac = 31^2 - 4 \times 1 \times 108 = 961 - 432 = 529$$ 5. **Find the roots:** $$t = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-31 \pm \sqrt{529}}{2} = \frac{-31 \pm 23}{2}$$ 6. **Calculate each root:** $$t_1 = \frac{-31 + 23}{2} = \frac{-8}{2} = -4$$ $$t_2 = \frac{-31 - 23}{2} = \frac{-54}{2} = -27$$ 7. **Answer:** The two numbers are **-4 and -27**. These numbers multiply to 108 and add to -31 as required.