$$\\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}+\\frac{6}…

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

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

Answer

Explanation:

Step1: Simplify the top numerator

$$ \frac{x+1}{x^2-1} + \frac{2}{x-1} = \frac{x+1}{(x-1)(x+1)} + \frac{2}{x-1} = \frac{1}{x-1} + \frac{2}{x-1} = \frac{3}{x-1} $$

Step2: Simplify the top denominator

$$ \frac{3}{x+1} - \frac{4}{x^2-4} = \frac{3}{x+1} - \frac{4}{(x-2)(x+2)} = \frac{3(x^2-4) - 4(x+1)}{(x+1)(x^2-4)} = \frac{3x^2-4x-16}{(x+1)(x-2)(x+2)} $$

Step3: Simplify the main numerator

$$ \frac{\frac{3}{x-1}}{\frac{3x^2-4x-16}{(x+1)(x-2)(x+2)}} = \frac{3(x+1)(x-2)(x+2)}{(x-1)(3x^2-4x-16)} $$

Step4: Simplify the main denominator

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

Step5: Divide main numerator by denominator

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

Step6: Combine and simplify terms

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

Answer:

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