Eliminate $n$. From the linear equation, $n=4\ell +m$. Substitute into the second:

$2m(4\ell +m)+10\ell (4\ell +m)+3\ell m=0$

$\Rightarrow 40{\ell }^2+21\ell m+2m^2=0\Rightarrow (8\ell +m)(5\ell +2m)=0$

Case 1: $m=-8\ell$, then $n=4\ell -8\ell =-4\ell$. Direction ratios $⟨1,-8,-4⟩$.

Case 2: $m=-\frac{5\ell }{2}$, then $n=4\ell -\frac{5\ell }{2}=\frac{3\ell }{2}$. Direction ratios $⟨2,-5,3⟩$.

Cosine of angle.

$\cos \theta =\frac{|(1)(2)+(-8)(-5)+(-4)(3)|}{\sqrt{1+64+16}\sqrt{4+25+9}}=\frac{|2+40-12|}{\sqrt{81}\cdot \sqrt{38}}=\frac{30}{9\sqrt{38}}=\frac{10}{3\sqrt{38}}$