You are given the equation:

$5x - 6 - x^2 = 0$

This is a quadratic equation, and we can solve it step by step:

Step 1: Rearrange the equation into standard form

We want to write the equation in the standard quadratic form: $ax^2 + bx + c = 0$. Rearranging terms, we get:

$-x^2 + 5x - 6 = 0$

or, multiplying by $-1$ to make the coefficient of $x^2$ positive:

$x^2 - 5x + 6 = 0$

Step 2: Factor the quadratic equation

Now, we factor the quadratic expression on the left-hand side. We need to find two numbers that multiply to give 6 and add to give $-5$. These numbers are $-2$ and $-3$.

So, we can factor the equation as:

$(x - 2)(x - 3) = 0$

Step 3: Solve for $x$

Set each factor equal to 0:

1. $x - 2 = 0$ ⟹ $x = 2$
2. $x - 3 = 0$ ⟹ $x = 3$

Final solution:

The solutions to the equation are:

$x = 2 \quad \text{or} \quad x = 3$

Would you like more details or have any questions? Here are 5 related questions to expand this concept:

1. How do you apply the quadratic formula to solve equations that can't be factored?
2. Can every quadratic equation be factored, or are there exceptions?
3. What are the different methods to solve quadratic equations (e.g., factoring, completing the square)?
4. How would the solution change if there was a positive constant on the right side of the equation?
5. How can you determine the nature of the solutions (real or complex) without solving the equation?

Tip: If the quadratic equation cannot be easily factored, the quadratic formula is a reliable method to find the solutions.