[tex]x;\ y-the\ numbers\\\\ \left\{\begin{array}{ccc}xy=-20\\x+y=19&\Rightarrow&x=19-y\end{array}\right\\\\subtitute\ to\ first\ equation:\\\\(19-y)y=-20\\19y-y^2+20=0\\-y^2+19y+20=0\ \ \ |change\ signs\\y^2-19y-20=0\\y^2-20y+y-20=0\\y(y-20)+1(y-20)+0\\(y-20)(y+1)=0\iff y-20=0\ or\ y+1=0\\\\y=20\ or\ y=-1[/tex]
[tex]x=19-y\\\\for\ y=20\Rightarrow x=19-20=-1\\\\for\ y=-1\Rightarrow x=19-(-1)=20\\\\Answer:\boxed{\boxed{20\ and\ -1}}[/tex]
