if

Given : $\left[\begin{array}{lll}x & 4 & 1\end{array}\right]\left[\begin{array}{ccc}2 & 1 & 2 \\ 1 & 0 & 2 \\ 0 & 2 & -4\end{array}\right]\left[\begin{array}{c}x \\ 4 \\ -1\end{array}\right]=0$

$\Rightarrow\left[\begin{array}{lll}2 x+4+0 & x+0+2 & 2 x+8-4\end{array}\right]\left[\begin{array}{c}x \\ 4 \\ -1\end{array}\right]=0$

$\Rightarrow\left[\begin{array}{lll}2 x+4 & x+2 & 2 x+4\end{array}\right]\left[\begin{array}{c}x \\ 4 \\ -1\end{array}\right]=0$

$\Rightarrow\left[2 x^{2}+4 x+4 x+8-2 x-4\right]=0$

$\Rightarrow 2 x^{2}+6 x+4=0$

$\Rightarrow x^{2}+3 x+2=0$

$\Rightarrow x^{2}+2 x+x+2=0$

$\Rightarrow x(x+2)+1(x+2)=0$

$\Rightarrow(x+2)(x+1)=0$

$\Rightarrow x+2=0$ or $x+1=0$

$\Rightarrow x=-2$ or $x=-1$

$\therefore x=-2,-1$