Let's solve the equation by factoring step by step.

Part 1: Rewrite the equation in factored form

The given equation is: $x^2 = 5x + 14$

First, let's move all terms to one side of the equation to set it equal to zero: $x^2 - 5x - 14 = 0$

Next, we need to factor this quadratic equation. We are looking for two numbers that multiply to $-14$ (the constant term) and add to $-5$ (the coefficient of the $x$ term). These numbers are $-7$ and $2$, so we can factor the quadratic as: $(x - 7)(x + 2) = 0$

Part 2: The solution set

To find the solution set, we set each factor equal to zero: $x - 7 = 0 \quad \text{or} \quad x + 2 = 0$

Solving these equations gives: $x = 7 \quad \text{or} \quad x = -2$

Therefore, the solution set is: $\{7, -2\}$

Would you like more details on how this factoring process works, or do you have any other questions?

Here are 5 related questions to expand your understanding:

1. How would you factor a quadratic equation if the coefficient of $x^2$ is not 1?
2. Can you explain how the quadratic formula relates to factoring?
3. What happens if a quadratic equation cannot be factored using integers?
4. How do you determine if a quadratic equation has real or complex solutions?
5. What are some methods to check if your factored form is correct?

Tip: When factoring quadratics, always double-check by expanding your factors to ensure they match the original equation.