\(\left(3k+1\right)\left(4k+2\right)\left(5k+3\right)=192\)
\(\Leftrightarrow60k^3+36k^2+30k^2+18k+20k^2+12k+10k+6=192\)
\(\Leftrightarrow60k^3+86k^2+40k-186=0\)
\(\Leftrightarrow2\left(30k^2+73k+93\ne0\right)\left(k-1\right)=0\)
\(\Leftrightarrow k=1\)