Answer
The factor of the given expression $\frac{\frac{{{x}^{2}}}{\sqrt{{{x}^{2}}+2}}-\sqrt{{{x}^{2}}+2}}{{{x}^{2}}}$ is $-\frac{2\sqrt{{{x}^{2}}+2}}{{{x}^{4}}+2{{x}^{2}}}$ .
Work Step by Step
Consider the expression: $\frac{\frac{{{x}^{2}}}{\sqrt{{{x}^{2}}+2}}-\sqrt{{{x}^{2}}+2}}{{{x}^{2}}}$
Multiply and divide the second term of the numerator $\sqrt{{{x}^{2}}+2}$
$\frac{\frac{{{x}^{2}}}{\sqrt{{{x}^{2}}+2}}-\sqrt{{{x}^{2}}+2}}{{{x}^{2}}}=\frac{\frac{{{x}^{2}}}{\sqrt{{{x}^{2}}+2}}-\frac{\sqrt{{{x}^{2}}+2}\sqrt{{{x}^{2}}+2}}{\sqrt{{{x}^{2}}+2}}}{{{x}^{2}}}$
Apply the radical rule: $\sqrt{a}\cdot \sqrt{a}=a$ and take the lowest common multiple
$\frac{\frac{{{x}^{2}}}{\sqrt{{{x}^{2}}+2}}-\frac{\sqrt{{{x}^{2}}+2}\sqrt{{{x}^{2}}+2}}{\sqrt{{{x}^{2}}+2}}}{{{x}^{2}}}=\frac{\frac{{{x}^{2}}-\left( {{x}^{2}}+2 \right)}{\sqrt{{{x}^{2}}+2}}}{{{x}^{2}}}$
Expand the numerator
$\begin{align}
& \frac{\frac{{{x}^{2}}-\left( {{x}^{2}}+2 \right)}{\sqrt{{{x}^{2}}+2}}}{{{x}^{2}}}=\frac{\frac{{{x}^{2}}-{{x}^{2}}-2}{\sqrt{{{x}^{2}}+2}}}{{{x}^{2}}} \\
& =\frac{\frac{-2}{\sqrt{{{x}^{2}}+2}}}{{{x}^{2}}}
\end{align}$
Apply the fraction rule: $\frac{\frac{b}{c}}{a}=\frac{b}{c\cdot a}$
$\frac{\frac{-2}{\sqrt{{{x}^{2}}+2}}}{{{x}^{2}}}=-\frac{2}{{{x}^{2}}\sqrt{{{x}^{2}}+2}}$
Rationalize the denominator by multiplying the numerator and denominator by $\frac{\sqrt{{{x}^{2}}+2}}{\sqrt{{{x}^{2}}+2}}$ .
$-\frac{2}{{{x}^{2}}\sqrt{{{x}^{2}}+2}}=-\frac{2\sqrt{{{x}^{2}}+2}}{{{x}^{2}}\sqrt{{{x}^{2}}+2}\times \sqrt{{{x}^{2}}+2}}$
Apply the radical rule: $\sqrt{a}\cdot \sqrt{a}=a$
$-\frac{2\sqrt{{{x}^{2}}+2}}{{{x}^{2}}\sqrt{{{x}^{2}}+2}\times \sqrt{{{x}^{2}}+2}}=-\frac{2\sqrt{{{x}^{2}}+2}}{{{x}^{2}}\left( {{x}^{2}}+2 \right)}$
Expand the denominator
$-\frac{2\sqrt{{{x}^{2}}+2}}{{{x}^{2}}\left( {{x}^{2}}+2 \right)}=-\frac{2\sqrt{{{x}^{2}}+2}}{{{x}^{2}}\cdot {{x}^{2}}+2{{x}^{2}}}$
Apply the exponent rule: ${{a}^{b}}\cdot {{a}^{c}}={{a}^{b+c}}$
$-\frac{2\sqrt{{{x}^{2}}+2}}{{{x}^{2}}\cdot {{x}^{2}}+2{{x}^{2}}}=-\frac{2\sqrt{{{x}^{2}}+2}}{{{x}^{4}}+2{{x}^{2}}}$
The simplified form of the expression $\frac{\frac{{{x}^{2}}}{\sqrt{{{x}^{2}}+2}}-\sqrt{{{x}^{2}}+2}}{{{x}^{2}}}$ is $-\frac{2\sqrt{{{x}^{2}}+2}}{{{x}^{4}}+2{{x}^{2}}}$ .
