The expression given is:

\[
\frac{24x - 12}{x - 2x^2}
\]

### Step 1: Simplify the expression.
First, factor out any common terms from the numerator and the denominator.

#### Numerator:
Factor out the greatest common factor (GCF) from \(24x - 12\):
\[
24x - 12 = 12(2x - 1)
\]

#### Denominator:
Factor out the greatest common factor (GCF) from \(x - 2x^2\):
\[
x - 2x^2 = x(1 - 2x)
\]

### Step 2: Rewrite the expression.
The expression becomes:
\[
\frac{12(2x - 1)}{x(1 - 2x)}
\]

### Step 3: Simplify the fraction.
Notice that \(2x - 1\) in the numerator is the negative of \(1 - 2x\) in the denominator:
\[
2x - 1 = -(1 - 2x)
\]
So we can simplify as follows:
\[
\frac{12(2x - 1)}{x(1 - 2x)} = \frac{12[-(1 - 2x)]}{x(1 - 2x)} = \frac{-12(1 - 2x)}{x(1 - 2x)} = \frac{-12}{x}
\]

### Final Answer:
The simplified expression is:
\[
\frac{-12}{x}
\]

Would you like more details or have any questions? Here are some related questions to explore further:

1. How would you handle simplifying the expression if the terms did not have a common factor?
2. What is the domain of the simplified function \(\frac{-12}{x}\)?
3. How do you interpret the behavior of the function as \(x\) approaches zero?
4. Can the original expression \(\frac{24x - 12}{x - 2x^2}\) be simplified further under any conditions?
5. How would the expression change if there was a constant added to the numerator?

**Tip:** Always check for common factors and potential simplifications before attempting more complex algebraic manipulations.

Learn how to simplify the rational expression (24x - 12) / (x - 2x^2) step-by-step. This problem involves algebraic manipulation and factoring of rational expressions. Suitable for high school students.

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