$$\\frac{\\frac{\\frac{x+1}{x^2-1} + \\frac{2}{x-1}}{\\frac{3}{x+1} - \\frac{4}{x^2-4}}}{\\frac{5}{x-2} +…

Explanation:

Step1: Simplify the top-most numerator

Factor $x^2-1$ as $(x-1)(x+1)$ and simplify the first term. $$ \frac{x+1}{(x-1)(x+1)} + \frac{2}{x-1} = \frac{1}{x-1} + \frac{2}{x-1} = \frac{3}{x-1} $$

Step2: Simplify the middle numerator

Find a common denominator for the middle terms. $$ \frac{3}{x+1} - \frac{4}{(x-2)(x+2)} = \frac{3(x^2-4) - 4(x+1)}{(x+1)(x^2-4)} = \frac{3x^2 - 4x - 16}{(x+1)(x^2-4)} $$

Step3: Simplify the bottom denominator

Find a common denominator for the bottom terms. $$ \frac{5}{x-2} + \frac{6}{(x-3)(x+3)} = \frac{5(x^2-9) + 6(x-2)}{(x-2)(x^2-9)} = \frac{5x^2 + 6x - 57}{(x-2)(x^2-9)} $$

Step4: Combine the top and middle parts

Divide the result of Step 1 by the result of Step 2. $$ \frac{\frac{3}{x-1}}{\frac{3x^2-4x-16}{(x+1)(x-2)(x+2)}} = \frac{3(x+1)(x-2)(x+2)}{(x-1)(3x^2-4x-16)} $$

Step5: Divide by the bottom denominator

Multiply the result of Step 4 by the reciprocal of Step 3. $$ \frac{3(x+1)(x-2)(x+2)}{(x-1)(3x^2-4x-16)} \cdot \frac{(x-2)(x-3)(x+3)}{5x^2+6x-57} $$

Step6: Final expression assembly

Combine all factors into a single rational expression. $$ \frac{3(x+1)(x-2)^2(x+2)(x-3)(x+3)}{(x-1)(3x^2-4x-16)(5x^2+6x-57)} $$

Answer:

$$ \frac{3(x+1)(x-2)^2(x+2)(x^2-9)}{(x-1)(3x^2-4x-16)(5x^2+6x-57)} $$