If $x=\frac{5-\sqrt{21}}{2}$, then $\left(x^3+\frac{1}{x^3}\right)-5\left(x^2+\frac{1}{x^2}\right)+\left(x+\frac{1}{x}\right)$ is

$x=\frac{5-\sqrt{21}}{2}$, $\frac{1}{x}=\frac{5+\sqrt{21}}{2}$

$(x^3+\frac{1}{x^3})-5(x^2+\frac{1}{x^2})+(x+\frac{1}{x})$

$=(x+\frac{1}{x})(x^2+\frac{1}{x^2}-1)-5({(x+\frac{1}{x})}^2-2)+(x+\frac{1}{x})$

$=(x+\frac{1}{x})({(x+\frac{1}{x})}^2-3)-5({(x+\frac{1}{x})}^2-2)+(x+\frac{1}{x})$

$x+\frac{1}{x}=5$

$=5(25-3)-5(25-2)+5$

$=110-115+5$

$=0$