The expression in the image is complex, and I'll break it down step-by-step for a clearer understanding. The expression appears as:

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

Let me simplify this systematically.

First part: Simplification of the first fraction

$\frac{x - \frac{4}{x}}{\frac{4}{x^2} - 9}$

Step 1: Simplify the numerator $x - \frac{4}{x}$

We can rewrite the numerator as:

$\frac{x^2 - 4}{x}$

Step 2: Simplify the denominator $\frac{4}{x^2} - 9$

Notice that the expression can be written as:

$\frac{4 - 9x^2}{x^2} = \frac{-9x^2 + 4}{x^2}$

Thus, the first fraction becomes:

$\frac{\frac{x^2 - 4}{x}}{\frac{4 - 9x^2}{x^2}}$

Simplifying this by multiplying the numerator by the reciprocal of the denominator:

$\frac{x^2 - 4}{x} \times \frac{x^2}{4 - 9x^2} = \frac{x(x^2 - 4)}{4 - 9x^2}$

Recognizing that $x^2 - 4 = (x - 2)(x + 2)$, the expression becomes:

$\frac{x(x - 2)(x + 2)}{4 - 9x^2}$

Since $4 - 9x^2 = -(9x^2 - 4) = -(3x + 2)(3x - 2)$, the first part simplifies to:

$-\frac{x(x - 2)(x + 2)}{(3x - 2)(3x + 2)}$

Second part: Simplification of the second fraction

$\frac{2}{x + 2} + \frac{1}{x - 2}$

To add these fractions, we find a common denominator:

$\frac{2}{x + 2} + \frac{1}{x - 2} = \frac{2(x - 2) + 1(x + 2)}{(x + 2)(x - 2)}$

Simplifying the numerator:

$\frac{2(x - 2) + (x + 2)}{(x + 2)(x - 2)} = \frac{2x - 4 + x + 2}{(x^2 - 4)} = \frac{3x - 2}{x^2 - 4}$

Combining both parts

We now multiply the two simplified parts:

$-\frac{x(x - 2)(x + 2)}{(3x - 2)(3x + 2)} \times \frac{3x - 2}{x^2 - 4}$

Cancelling out common terms where possible, we find that $(x - 2)(x + 2) = x^2 - 4$, so we can cancel those terms from the numerator and denominator:

$-\frac{x \cdot (3x - 2)}{(3x - 2)(3x + 2)}$

Further canceling $3x - 2$ terms, the final expression becomes:

$-\frac{x}{3x + 2}$

Would you like further clarification on any of the steps?

5 Related Questions:

1. How can you handle similar rational expressions involving multiple variables?
2. What is the significance of factoring in simplifying algebraic fractions?
3. How does recognizing common terms help in simplifying algebraic expressions?
4. What methods can be used to combine fractions with different denominators?
5. How would the result change if certain terms in the expression were altered?

Tip:

When simplifying complex fractions, always look for opportunities to factor both numerators and denominators. This often reveals common terms that can be canceled out.