To create an informative article on solving the given mathematical problems, I’ll break down the process into clear steps with headings for each equation. Let’s start with the first equation:

## 1. Solving the Linear Equation: 5x – 12 = 0

#### Step 1: Set Up the Equation

The given equation is a linear equation in one variable, which is **5x – 12 = 0**.

#### Step 2: Isolate the Variable

To find the value of x, we need to isolate it. This can be done by adding 12 to both sides of the equation:

[ 5x – 12 + 12 = 0 + 12 ]

Simplifying this, we get:

[ 5x = 12 ]

#### Step 3: Solve for x

Now, divide both sides by 5:

[ x = \frac{12}{5} ]

So, the solution is ( x = \frac{12}{5} ) or ( x = 2.4 ).

## 2. Understanding the Expression: 4x^2 – 5x

This is not an equation but an algebraic expression, as it doesn’t have an equals sign. Therefore, it cannot be solved but can be simplified or factored if possible. The expression is already in its simplest form.

## 3. Solving the Linear Equation: -5x – 12 = 0

This equation is similar to the first one.

#### Step 1: Set Up the Equation

The equation is -5x – 12 = 0.

#### Step 2: Isolate the Variable

Add 12 to both sides:

[ -5x – 12 + 12 = 0 + 12 ]

Simplifying this, we get:

[ -5x = 12 ]

#### Step 3: Solve for x

Now, divide both sides by -5:

[ x = \frac{12}{-5} ]

So, the solution is ( x = \frac{-12}{5} ) or ( x = -2.4 ).

## 4. Solving the Quadratic Equation: 4y^2 – 5x – 12 = 0

This appears to be a quadratic equation, but it contains two different variables (y and x). This is unusual for standard quadratic equations. If it’s meant to be in terms of y, we can solve it assuming x is a constant.

#### Step 1: Set Up the Equation

Assuming x is a constant, the equation is 4y^2 – 5x – 12 = 0.

#### Step 2: Solve for y

To solve for y, we would typically use the quadratic formula, but here it’s not possible without a specific value for x.

## 5. Understanding the Expression: -2 – 5x – 12

Like the second item, this is an algebraic expression and not an equation. It can be simplified by combining like terms:

[ -2 – 5x – 12 ]

[ = -5x – 14 ]

This expression is now in its simplest form.

For more accurate solutions, especially for the fourth problem, the context or additional information about the variables is needed.