When the centre of the ellipse is at (h,k), then the standard equation of the ellipse is:
((x-h)^2)/a^2+((y-k)^2)/b^2=1, where a and b are the lengths of the semi-major and semi-minor axes. The foci lie along the major axis, which must be at y=-6, since -6 is the shared coord for the two foci. The average of the x coords for the foci is (2+8)/2=5. This indicates where the centre of the ellipse is: (5,-6) so h=5 and k=-6. The distance f between the centre of the ellipse and the foci is given by f^2=a^2-b^2, so a^2-b^2=(5-2)^2=(8-5)^2=9. The major axis has length 10, so ther semi-major axis, a, has length 5. Therefore we can find b: 25-b^2=9, so b^2=16, b=4.
The equation of the ellipse is (x-5)^2/25+(y+6)^2/16=1.