Answer
$x_1=-4$ and $x_2=0$.
Work Step by Step
The system can written on the form $Ax=b$ as follows
$$\left[ \begin {array}{cc} {2}& {1}\\{1}&{4} \end {array} \right]\left[ \begin {array}{cc} {x_1}\\ {x_2} \end {array} \right]=\left[ \begin {array}{cc} {-8}\\ {-4} \end {array} \right].$$
To solve system using Gaussian elimination, we form the augmented matrix as follows
$$\left[ \begin {array}{ccc} 2&1&-8\\ 1&4&-4
\end {array} \right].
$$
The reduced row echelon form is given by
$$\left[ \begin {array}{ccc} 1&0&-4\\ 0&1&0
\end {array} \right].
$$
From which, we have $x_1=-4$ and $x_2=0$.
