Question 444190
We complete the square for x and y on the left side by adding 28+12 on both sides:

{{{7x^2-28x+3y^2-12y=-19}}}=> {{{7(x^2-4x+4)*3(y^2-4y+4)=-19+28+12}}}=>

{{{7(x-2)^2+3(y-2)^2=21}}}, divide both sides by 21

{{{(x-2)^2/3+(y-2)^2/7=1}}}, that is the equation of ellipse with center (2, 2),

major axis {{{2sqrt(7)}}}, minor axis {{{2sqrt(3)}}}, 

and foci [ {{{c^2=7^2-3^2=40}}} => {{{c=2sqrt(10)}}}] the points:

(2, {{{2+2sqrt(10)}}}) and (2, {{{2-2sqrt(10)}}}).

{{{drawing(300, 300, -5, 5, -5, 5, grid(1), ellipse( 2, 2, 2sqrt(3), 2sqrt(7)))}}}.