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Denominator:
\frac{2x^4 + 2}{x^4 - 1} = 2 \cdot \frac{x^4 + 1}{x^4 - 1}
Let’s simplify $ x^4 + 1 $ and $ x^4 - 1 $. Since $ x + \frac{1}{x} = 5 $, square again:
x^2 + \frac{1}{x^2} = 23 \Rightarrow x^4 + \frac{1}{x^4} = (x^2 + \frac{1}{x^2})^2 - 2 = 23^2 - 2 = 529 - 2 = 527
x^4 + 1 = x^4 + \frac{1}{x^4} \cdot x^4 = x^4 + \frac{x^4}{x^4} = x^4 + 1 \Rightarrow \text{Not directly helpful}
Instead, observe:
\frac{x^2 + 1}{x^2 - 1} + \frac{x^2 - 1}{x^2 + 1} = y + \frac{1}{y} = \frac{x^2 + 1}{x^2 - 1} + \frac{x^2 - 1}{x^2 + 1}
Let’s compute this directly using the identity:
\frac{a + b}{a - b} + \frac{a - b}{a + b} = \frac{(a + b)^2 + (a - b)^2}{(a - b)(a + b)} = \frac{2a^2 + 2b^2}{a^2 - b^2}
\frac{2(x^2 + 1)}{x^2 - 1}