To factor and solve the quadratic equation $x^2 - 4x - 5 = 0$, let's follow these steps:

1. Find two numbers that multiply to give $-5$ and add to give $-4$. These numbers are $-5$ and $1$, because: $(-5) \times 1 = -5 \quad \text{and} \quad (-5) + 1 = -4.$

2. Rewrite the quadratic as: $(x - 5)(x + 1) = 0.$

3. Solve for $x$ by setting each factor equal to zero: $x - 5 = 0 \quad \text{or} \quad x + 1 = 0.$ This gives: $x = 5 \quad \text{or} \quad x = -1.$

Thus, the exact solutions are: $x = 5; -1.$

Would you like any further details or explanations?

Here are five related questions:

1. How do you verify the solutions of a quadratic equation?
2. What are the other methods to solve a quadratic equation besides factoring?
3. How can the quadratic formula be derived from the standard form of a quadratic equation?
4. What happens if the discriminant of a quadratic equation is negative?
5. Can all quadratic equations be factored?

Tip: Always check if factoring is possible by finding two numbers that satisfy the product-sum rule. If not, use the quadratic formula!