Parametrise the first line: $P=(2\lambda +1,-3\lambda -1,\lambda )$. Distance from $(1,-1,0)$: $\sqrt{(2\lambda {)}^2+(-3\lambda {)}^2+{\lambda }^2}=\sqrt{14}|\lambda |=4\sqrt{14}$, so $|\lambda |=4$.

For the point nearer the origin, take $\lambda =-4$, giving $P=(-7,11,-4)$.

Line through $P$ has direction $(1,2,3)$; the other line passes through $(-5,10,3)$ with direction $(2,1,1)$.

Vector between points: $(-5-(-7),10-11,3-(-4))=(2,-1,7)$.

${\vec{d}}_1\times {\vec{d}}_2=\left|\begin{array}{c}\hat{i}\hat{j}\hat{k}\\ 123\\ 211\end{array}\right|=-\hat{i}+5\hat{j}-3\hat{k}$, magnitude $\sqrt{1+25+9}=\sqrt{35}$.

Numerator: $|(2)(-1)+(-1)(5)+(7)(-3)|=28$.

Shortest distance $=\frac{28}{\sqrt{35}}=\frac{4\sqrt{35}}{5}=4\sqrt{\frac{7}{5}}$.