1. **State the problem:** Find all values of $n$ for which the quadratic equation $$(n + 1)s^2 - 4s - 6 = 0$$ has exactly one real solution. 2. **Recall the condition for one real solution:** A quadratic equation $ax^2 + bx + c = 0$ has exactly one real solution if its discriminant is zero. 3. **Write the discriminant formula:** $$\Delta = b^2 - 4ac$$ 4. **Identify coefficients:** $$a = n + 1, \quad b = -4, \quad c = -6$$ 5. **Set discriminant to zero:** $$(-4)^2 - 4(n + 1)(-6) = 0$$ 6. **Simplify:** $$16 + 24(n + 1) = 0$$ 7. **Distribute:** $$16 + 24n + 24 = 0$$ 8. **Combine like terms:** $$24n + 40 = 0$$ 9. **Isolate $n$:** $$24n = -40$$ 10. **Simplify fraction:** $$n = \frac{-40}{24} = \frac{\cancel{-40}}{\cancel{24}} = \frac{-10}{6} = \frac{-5}{3}$$ **Final answer:** $$n = -\frac{5}{3}$$