https://file.aiquickdraw.com/mathbot/file/1726113335481kzxkvl82.jpg

The given expression in the image is:

\[
\frac{\left( \frac{x - 4}{x} \right)}{\frac{4}{x^2} - 9} \times \left( \frac{2}{x + 2} + \frac{1}{x - 2} \right)
\]

Let’s break this into two parts and simplify each:

### Part 1: Simplification of \(\frac{\left( \frac{x - 4}{x} \right)}{\frac{4}{x^2} - 9}\)

1. The denominator \(\frac{4}{x^2} - 9\) can be simplified using the difference of squares:
   \[
   \frac{4}{x^2} - 9 = \frac{4}{x^2} - \frac{9x^2}{x^2} = \frac{4 - 9x^2}{x^2}
   \]
   So, the denominator becomes:
   \[
   \frac{4 - 9x^2}{x^2}
   \]

   Now, the expression becomes:
   \[
   \frac{\frac{x - 4}{x}}{\frac{4 - 9x^2}{x^2}} = \frac{x - 4}{x} \times \frac{x^2}{4 - 9x^2}
   \]
   This simplifies to:
   \[
   \frac{(x - 4)x}{4 - 9x^2}
   \]

### Part 2: Simplification of \(\frac{2}{x + 2} + \frac{1}{x - 2}\)

1. The expression has two fractions, so we need a common denominator:
   \[
   \frac{2}{x + 2} + \frac{1}{x - 2} = \frac{2(x - 2) + 1(x + 2)}{(x + 2)(x - 2)}
   \]
   Simplify the numerator:
   \[
   2(x - 2) + 1(x + 2) = 2x - 4 + x + 2 = 3x - 2
   \]
   So, this becomes:
   \[
   \frac{3x - 2}{(x + 2)(x - 2)}
   \]

### Final Expression:

Combining the two simplified parts, we get:
\[
\frac{(x - 4)x}{4 - 9x^2} \times \frac{3x - 2}{(x + 2)(x - 2)}
\]

This is the simplified form of the given expression.

Would you like further details on this or any additional explanations?

### 5 Related Questions:
1. How do you solve for \(x\) when there are fractions in the equation?
2. Can we factor further expressions like \(4 - 9x^2\) into more specific terms?
3. How do we deal with multiple terms in the denominator when simplifying complex fractions?
4. What is the difference of squares, and how is it used in algebraic expressions?
5. How would the expression change if constants were added inside the fractions?

### Tip:
Always check for common factors before multiplying complex fractions—it simplifies the process and reduces errors.

Simplify the given expression: ( (x - 4) / x ) / ( (4 / x^2) - 9 ) * ( (2 / (x + 2)) + (1 / (x - 2)) )

Learn how to simplify a complex algebraic expression involving fractions and the difference of squares. The solution provides a step-by-step breakdown of how to approach the problem, focusing on fraction operations and algebraic techniques. Suitable for students in Grades 10-12.

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