The major axis has length 10 along the x-axis nad is centered at (0,0), so its
endpoints are at (-5,0) nad (5,0).  Thus, a = 5.  Likewise, b = 3.  So the
equation of this ellipse is:
 x2      y2                x2      y2
---- + ---- = 1     or    ---- + ---- = 1
 52      32                25      9

Collect together the terms involving x and those involving y and complete the 
square (remember that you are looking for things like (x - h)2 and 
(y - k)2).

x2 + 9y2 - 4x - 72y + 139 = 0
(x2 - 4x) + 9(y2 - 8y) + 139 = 0
(x2 - 4x + 4) -4 + 9(y2 - 8y + 16) - 9(16) + 139 = 0
(x - 2)2 + 9(y - 4)2 = 9

(x - 2)2   (y - 4)2
-------- + -------- = 1
    9          1

So, (h, k) = (2, 4), a = 3 and b = 1.

Therefore, the equation represents an ellipse centered at (2,4) with major axis parallel to the x-axis of length 6 and minor axis parallel to the y-axis of length 2.

Proceed as before:
3x2 + 2y2 + 24x - 4y + 62 = 0
3(x2 + 8x) + 2(y2 - 2y) + 62 = 0
3(x2 + 8x + 16) - 3(16) + 2(y2 - 2y + 1) - 2(1) + 62 = 0
3(x + 4)2 + 2(y - 1)2 = -12

(x + 4)2    (y - 1)2
-------- + -------- = -2
    2         3

There are no points (x, y) which satisfy this equation.  The reason is that the 
left hand side is always greater than 0, no matter what x and y may be.  So the 
left hand side can never equal -2.  Thus, there is no curve represented by
3x2 + 2y2 + 24x - 4y + 62 = 0