To solve the cubic equation \(x^3 - 4x - 5 = 0\), we can attempt to use numerical methods or exploratory techniques to find its roots, as this equation does not easily lend itself to simple algebraic factorization. Here are some approaches you might consider:

### Graphical or Numerical Solution:
1. **Graphical Method**: Plot the function \(y = x^3 - 4x - 5\) and find the x-coordinates where the graph intersects the x-axis. These x-values are the real roots of the equation.

2. **Numerical Methods**: Use numerical techniques like the Newton-Raphson method or bisection method to find approximate values of the roots. With numerical methods or a calculator, you can find that there is one real root at approximately \(x \approx 2.14\).

### Analytical Solution:
Cubic equations can be solved using Cardano's formula, but it's often complicated for practical manual calculations. However, if the coefficients were simpler or had certain symmetries, it might be feasible to work through by hand.

### Factoring or Testing Rational Roots:
- Check possible rational roots using the Rational Root Theorem, which suggests that any potential rational root, expressed as a fraction \(\frac{p}{q}\), has \(p\) as a factor of the constant term (-5) and \(q\) as a factor of the leading coefficient (1). So, check potential roots \( \pm 1, \pm 5 \).
- Substitute these values into the equation to see if any yield zero. For this particular equation, neither \(\pm 1\) nor \(\pm 5\) are roots.

In summary, this equation has one real root, approximately \(x \approx 2.14\), and the other two roots are complex. For exact solutions or verification, numerical approximation tools or software would be best used to find all complex and real roots accurately.