https://file.aiquickdraw.com/mathbot/file/17254760660396d883gme.jpg

The expression in the image appears to be:

\[
\frac{10}{\frac{2x - 3x - 2}{x - 2} - 2}
\]

Let's simplify this step by step.

1. **Simplify the expression inside the fraction:**

   \[
   2x - 3x - 2 = -x - 2
   \]

2. **Substitute this into the expression:**

   \[
   \frac{10}{\frac{-x - 2}{x - 2} - 2}
   \]

3. **Simplify the fraction inside the denominator:**

   \[
   \frac{-x - 2}{x - 2} = \frac{-(x + 2)}{x - 2} = -\frac{x + 2}{x - 2}
   \]

4. **Substitute and simplify further:**

   \[
   \frac{10}{-\frac{x + 2}{x - 2} - 2}
   \]

5. **Combine the terms in the denominator:**

   To combine the terms in the denominator, get a common denominator:

   \[
   -\frac{x + 2}{x - 2} - 2 = -\frac{x + 2}{x - 2} - \frac{2(x - 2)}{x - 2} = \frac{-(x + 2) - 2(x - 2)}{x - 2}
   \]

   Simplify the expression inside the numerator:

   \[
   -(x + 2) - 2(x - 2) = -x - 2 - 2x + 4 = -3x + 2
   \]

   So, the expression becomes:

   \[
   \frac{10}{\frac{-3x + 2}{x - 2}} = 10 \times \frac{x - 2}{-3x + 2} = \frac{10(x - 2)}{-3x + 2}
   \]

This is the simplified form of the given expression:

\[
\boxed{\frac{10(x - 2)}{-3x + 2}}
\]

Would you like more details or have any other questions?

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Here are five related questions for further exploration:

1. How would the result change if the original expression had different coefficients?
2. Can this expression be factored further if specific values of \( x \) are substituted?
3. What happens to the expression as \( x \) approaches the value that makes the denominator zero?
4. How would the approach change if the expression was a polynomial of higher degree?
5. What is the general method for simplifying complex rational expressions like this?

**Tip:** Always check if the expression can be further simplified by factoring both the numerator and denominator.

Simplify the given mathematical expression.

Learn how to simplify a complex rational expression step by step, including simplification of fractions and factoring techniques. This example is ideal for students in Grades 10-12 who are studying advanced algebra.

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