\(\displaystyle \frac{x^2+1}{x}+\frac{x}{x^2+1} \ = \ \frac{29}{10}, \ I'm \ assuming \ that \ you \ want \ to \ solve \ for \ x.\)

\(\displaystyle \frac{(x^2+1)^2+x^2}{x(x^2+1)} \ = \ \frac{29}{10}, \ \frac{x^4+3x^2+1}{x^3+x} \ = \ \frac{29}{10}, \ 10x^4+30x^2+10 \ = \ 29x^3+29x.\)

\(\displaystyle Descartes' \ Rule \ of \ Signs:\)

\(\displaystyle Let \ f(x) \ = \ 10x^4-29x^3+30x^2-29x+10, \ 4 \ changes.\)

\(\displaystyle f(-x) \ = \ 10x^4+29x^3+30x^2+29x+10, \ 0 \ changes.\)

\(\displaystyle Hence, \ three \ possibilities: \ (4+,0-,0I),(2+,0-,2I), \ and \ (0+,0-,4I).\)

\(\displaystyle Ergo, \ f(x) \ = \ (x-2)(2x-1)(5x^2-2x+5), \ second \ of \ the \ three \ possibilities.\)

\(\displaystyle Note: \ x \ = \ 2 \ and \ x \ = \ \frac{1}{2} \ in \ real \ number \ land, \ see \ graph.\)

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