If the system of equations
$\begin{array}{l}(\mathrm{\lambda }-1)\mathrm{x}+(\mathrm{\lambda }-4)\mathrm{y}+\mathrm{\lambda z}=5\\ \mathrm{\lambda x}+(\mathrm{\lambda }-1)\mathrm{y}+(\mathrm{\lambda }-4)\mathrm{z}=7\\ (\mathrm{\lambda }+1)\mathrm{x}+(\mathrm{\lambda }+2)\mathrm{y}-(\mathrm{\lambda }+2)\mathrm{z}=9\end{array}$
has infinitely many solutions, then $\mathrm{\lambda }$² + $\mathrm{\lambda }$ is equal

$\begin{array}{l}(\mathrm{\lambda }-1)\mathrm{x}+(\mathrm{\lambda }-4)\mathrm{y}+\mathrm{\lambda z}=5\\ \mathrm{\lambda x}+(\mathrm{\lambda }-1)\mathrm{y}+(\mathrm{\lambda }-4)\mathrm{z}=7\\ (\mathrm{\lambda }+1)\mathrm{x}+(\mathrm{\lambda }+2)\mathrm{y}-(\mathrm{\lambda }+2)\mathrm{z}=9\end{array}$
For infinitely many solutions
$\begin{array}{l}\mathrm{D}=\left|\begin{array}{ccc}\mathrm{\lambda }-1& \mathrm{\lambda }-4& \mathrm{\lambda }\\ \mathrm{\lambda }& \mathrm{\lambda }-1& \mathrm{\lambda }-4\\ \mathrm{\lambda }+1& \mathrm{\lambda }+2& -(\mathrm{\lambda }+2)\end{array}\right|=0\\ (\mathrm{\lambda }-3)(2\mathrm{\lambda }+1)=0\\ {\mathrm{D}}_{\mathrm{x}}=\left|\begin{array}{ccc}5& \mathrm{\lambda }-4& \mathrm{\lambda }\\ 7& \mathrm{\lambda }-1& \mathrm{\lambda }-4\\ 9& \mathrm{\lambda }+2& -(\mathrm{\lambda }+2)\end{array}\right|=0\\ 2(3-\mathrm{\lambda })(23-2\mathrm{\lambda })=0\\ \mathrm{\lambda }=3\\ \therefore {\mathrm{\lambda }}^2+\mathrm{\lambda }=9+3=12\end{array}$