The given system of homogeneous equations
$\begin{aligned} & t x+(t+1) y+(t-1) z=0 \\ & (t+1) x+t y+(t+2) z=0, \text { and } \\ & (t-1) x+(t+2) y+t z=0 \text { in } x, y, z \text { has a non-trivial }\end{aligned}$
solutions, then
$\Delta=0 \Rightarrow\left|\begin{array}{ccc}t & t+1 & t-1 \\ t+1 & t & t+2 \\ t-1 & t+2 & t\end{array}\right|=0$
On applying $R_2 \rightarrow R_2-R_1$ and $R_3 \rightarrow R_3-R_1$, we get
$\left|\begin{array}{ccc}t & t+1 & t-1 \\ 1 & -1 & 3 \\ -1 & 1 & 1\end{array}\right|=0$
$\begin{aligned} & \Rightarrow t(-1-3)-(t+1)(1+3)+(t-1)(1-1)=0 \\ & \Rightarrow-4 t-4 t-4=0 \Rightarrow t=-\frac{1}{2}\end{aligned}$
From the options $2 t^2+3 t+1=0$ satisfied at $t=-\frac{1}{2}$ Hence, option (c) is correct.