Technically speaking, you need to complete squares separately for x- and y- terms.


25x^2 + y^2 - 100x + 8y + 91 = 0     is equivalent to

(25x^2 - 100x) + (y^2 + 8y) = - 91   is equivalent to

25*(x^2 - 4x)  + (y^2 + 8y) = -91    is equivalent to

25*(x^2 - 4x + 4) + (y^2 + 8y + 16) = -91 + 100 + 16    is equivalent to

25*(x-2)^2 + (y+4)^2 = 25      (<<<---=== divide by 25 both sides ===--->>>)  is equivalent to

 +  = 1.


The last equation is the standard form equation of the ellipse 

   - centered at the point (2,-4)

   - with the major axis parallel to y-axis

   - taller than wide

   - with the major semi-axis of the length of 5 in vertical   direction and
          the minor semi-axis of the length of 1 in horizontal direction.


The linear eccentricity of the ellipse is   =  = .


The    vertices are  (2,-4+5) = (2,1)   and  (2,-4-5) = (2,-9).

The co-vertices are  (2+1,-4) = (3,-4)  and  (2-1,-4) = (1,-4).


The foci are  (2,-4+)  and  (2,-4-).


The plot is THIS:





 Ellipse  +  = 1