Transform the equation by completing the square:
4x^2 + y^2 - 8x + 6y -3 =0
Also show foci
4x^2 + y^2 - 8x + 6y -3 =0
4(x^2 - 2x) + (y^2 + 6y) = 3
4(x^2 - 2x + 1) + (y^2 + 6y + 9) = 3 + 4 + 9
4(x-1)^2 + (y+3)^2 = 16
(x-1)^2/2^2 + (y+3)^2/4^2 = 1
ellipse with center at (1,-3)
vertical major axis, of semi-length 4.
4^2 = c^2 + 2^2
c = 2√3
you should be able to place the foci from that.