Polynomial Root Manipulation

Since $x_1, x_2,x_3$ are the roots of given equation

By vieta's relations it follows that

$x_1+x_2+x_3=9$

$x_1x_2+x_2x_3+x_3x_1=11$

$x_1x_2x_3=1$

Now given $y= \sqrt{x_1}+ \sqrt{x_2}+ \sqrt{x_3}$

Squaring both sides we obtain

$y^2=x_1+x_2+x_3+2(\sqrt{x_1x_2}+\sqrt{x_2x_3}+\sqrt{x_3x_1})$

$\large y^2=9+2(\frac{1}{\sqrt{x_3}}+\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}})$

(Using vieta's relations above)

$\large (y^2-9)^2=4(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3})+2(\frac{1}{\sqrt{x_1x_2}}+\frac{1}{\sqrt{x_2x_3}}+\frac{1}{\sqrt{x_3x_1}}))$

$y^4-18y^2+81=4(\frac{x_1x_2+x_2x_3+x_3x_1}{x_1x_2x_3}+2(\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}))$

$y^4-18y^2+81=4(\frac{11}{1}+2y)$

$y^4-18y^2-8y=44-81=-37$

So $|y^4-18y^2-8y|=\boxed{37}$