Answer
No real solution.
Work Step by Step
$f\left( x \right)={{\left( \frac{{{x}^{2}}+2}{x} \right)}^{4}}+7{{\left( \frac{{{x}^{2}}+2}{x} \right)}^{2}}+5$.
The x-intercept occurs when $f\left( x \right)=0$,
${{\left( \frac{{{x}^{2}}+2}{x} \right)}^{4}}+7{{\left( \frac{{{x}^{2}}+2}{x} \right)}^{2}}+5=0$ …… (1)
Let $u={{\left( \frac{{{x}^{2}}+2}{x} \right)}^{2}}$ and ${{u}^{2}}={{\left( \frac{{{x}^{2}}+2}{x} \right)}^{4}}$,
Substitute the values of $u$ and ${{u}^{2}}$ in equation (1),
${{u}^{2}}+7u+5=0$
$\begin{align}
& x=\frac{-\left( 7 \right)\pm \sqrt{{{\left( 7 \right)}^{2}}-4\left( 1 \right)\left( 5 \right)}}{2\left( 1 \right)} \\
& =\frac{-7\pm \sqrt{49-20}}{2} \\
& =\frac{3\pm \sqrt{-29}}{2}
\end{align}$
So, there are no real solutions.