Between the Roots

To solve x^2 - 4x - 5 < 0, factor first: (x - 5)(x + 1) < 0. Now the question is about signs. A product of two numbers is negative exactly when one factor is positive and the other is negative, so there are two sign cases.

Case one: x - 5 > 0 and x + 1 < 0, so x is above 5 and below -1 at once. No number does both, so this case is empty. Case two: x - 5 < 0 and x + 1 > 0, so x is below 5 and above -1. That holds for every number between the roots, written as one chain: -1 < x < 5.

The answer is a whole stretch of the number line, not a list of numbers. Check one inside it: at x = 0 the factors are -5 and 1, and their product is negative, as asked. The roots themselves are out, since there the product is 0, not less than 0.