If the system of equations $2\mathrm{x}+3\mathrm{ky}+(3\mathrm{k}+4)\mathrm{z}=0,\mathrm{x}+(\mathrm{k}+4)\mathrm{y}+(4\mathrm{k}+2)\mathrm{z}=0$, $\mathrm{x}+2(\mathrm{k}+4)\mathrm{y}+(3\mathrm{k}+4)\mathrm{z}=0$has non trivial solution then k=

det A = 0 
$$\begin{gathered} \left|\begin{array}{ccc}2& 3k& 3k+4\\ 1& k+4& 4k+2\\ 1& 2(k+4)& 3k+4\end{array}\right|=0 \\ {R}_2\to 2R_2-R_{1,}{R}_3\to 2R_3-R_1 \\ \left|\begin{array}{ccc}2& 3k& 3k+4\\ 0& 8-k& 5k\\ 0& 16+k& 3k+4\end{array}\right|=0 \\ 2\left[(8-k)(3k+4)-5k(16+k)\right]=0 \\ 24k-4k-3k^2+32-80k-5k^2=0 \\ 8k^2+60k-32=0 \end{gathered}$$
 $$\begin{gathered} \Rightarrow 2{\mathrm{k}}^2+15\mathrm{k}-8=0 \\ \Rightarrow (2k+1)(k+8)=0 \\ k=-8,-\frac{1}{2} \end{gathered}$$