The system of linear equations

$D=\left|\begin{array}{ccc}\lambda & 2 & 2 \\ 2 \lambda & 3 & 5 \\ 4 & \lambda & 6\end{array}\right|$

$D=\lambda^{2}+6 \lambda-16$

$D=(\lambda+8)(2-\lambda)$

For no solutions, $D=0$

$\Rightarrow \lambda=-8,2$

when $\lambda=2$

$D_{1}=\left|\begin{array}{ccc}5 & 2 & 2 \\ 8 & 3 & 5 \\ 10 & 2 & 6\end{array}\right|$

$=5[18-10]-2[48-50]+2(16-30]$

$=40+4-28 \neq 0$

There exist no solutions for $\lambda=2$