Given hyperbola,
$14 x^{2}+38\, x y+20 \,y^{2}+x-7 y-91=0\,\,\,...(i)$
On factorising $14 x^{2}+38\, x y+20\, y^{2}$, we get
$=(7 x+5 y)(2 x+4 y)$
One of the asymptote is $7 x+5 y-3=0$
Then, let other asymptote is $2 x+4 y+k=0$
So, on combining
$(7 x+5 y-3)(2 x+4 y+k)=0 \,\,\,\ldots(ii)$
On equating the coefficient of $x$ from Eqs. (i) and (ii), we get
$7 k-6= 1$
$ \Rightarrow k=1$
So, other asymptote is, $2 x+4 y+1=0 $.