Answer
$\left[\begin{array}{llllll}
1 & -5 & 2 & -2 & | & 4\\
0 & 1 & -3 & -1 & | & 0\\
0 & 15 & -4 & 5 & | & -6\\
0 & -19 & 12 & -6 & | & 13
\end{array}\right]$
Work Step by Step
The elements of the third row will be replaced with
$-3(1)+3=0,$
$-3(-5)+0=15,$
$-3(2)+2=-4,$
$-3(-2)+(-1)=5,$
$-3(4)+6=-6$, to become
$0\ \ 15\ \ -4\ \ 5\ \ |\ \ -6.$
The elements of the fourth row will be replaced with
$4(1)+(-4)=0,$
$4(-5)+1=-19,$
$4(2)+4=12,$
$4(-2)+2=-6,$
$4(4)+(-3)=13$ , to become
$0\ \ -19\ \ 12\ \ -6\ \ |\ \ 13$
resulting in
$\left[\begin{array}{llllll}
1 & -5 & 2 & -2 & | & 4\\
0 & 1 & -3 & -1 & | & 0\\
0 & 15 & -4 & 5 & | & -6\\
0 & -19 & 12 & -6 & | & 13
\end{array}\right]$
