It looks like you're trying to solve an equation that involves factoring. Your input seems to contain some repeated parts, so I'll interpret it as an equation you want to solve using factoring. I think the equation you have in mind is:

\[
z^2 - 12 = 4z
\]

Let's solve this step-by-step using factoring.

### Step 1: Rewrite the Equation

Start by rewriting the equation to set it equal to zero:

\[
z^2 - 4z - 12 = 0
\]

### Step 2: Factor the Quadratic Equation

To factor the quadratic equation \(z^2 - 4z - 12 = 0\), we need to find two numbers that multiply to \(-12\) (the constant term) and add up to \(-4\) (the coefficient of \(z\)).

The numbers that satisfy this are \(-6\) and \(2\), because:

\[
-6 \times 2 = -12 \quad \text{and} \quad -6 + 2 = -4
\]

So, we can factor the quadratic as:

\[
(z - 6)(z + 2) = 0
\]

### Step 3: Solve for \(z\)

Set each factor equal to zero and solve for \(z\):

1. \(z - 6 = 0\)
   
   \[
   z = 6
   \]

2. \(z + 2 = 0\)
   
   \[
   z = -2
   \]

### Solution

The solutions to the equation \(z^2 - 4z - 12 = 0\) are:

\[
z = 6 \quad \text{and} \quad z = -2
\]

These are the values of \(z\) that satisfy the original equation.

Would you like more details, or do you have any questions?

### Related Questions:

1. How do you factor trinomials where the leading coefficient is not 1, such as \(3z^2 - 5z - 2 = 0\)?
2. What is the quadratic formula, and how does it relate to factoring?
3. How can you factor expressions that involve higher powers, like \(z^3 - 3z^2 - 4z + 12 = 0\)?
4. What is the difference between factoring and completing the square?
5. How can factoring be used to solve polynomial equations of degree higher than 2?

### Tip:

Always check your factored forms by expanding them back out to ensure they equal the original expression.

Learn how to solve the quadratic equation z^2 - 4z - 12 = 0 using factoring with a detailed step-by-step solution. This problem covers key algebraic concepts, including quadratic equations and factoring techniques. Suitable for high school students.

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