\[ 25 x^{2}+49 y^{2}-200 x+294 y-384=0 \] : Find the endpoints of the major and minor axes of this ellipse.

Step 1 ：Rewrite the given equation in the standard form of an ellipse equation. The standard form of an ellipse equation is \[(x-h)^2/a^2 + (y-k)^2/b^2 = 1\] where (h,k) is the center of the ellipse, and a and b are the lengths of the semi-major and semi-minor axes respectively.

Step 2 ：Given equation is \[25x^2 - 200x + 49y^2 + 294y - 384 = 0\]

Step 3 ：Rearrange the terms to group the x's and y's together: \[25(x^2 - 8x) + 49(y^2 + 6y) = 384\]

Step 4 ：Complete the square for the x and y terms: \[25[(x - 4)^2 - 16] + 49[(y + 3)^2 - 9] = 384\]

Step 5 ：Simplify: \[25(x - 4)^2 + 49(y + 3)^2 = 384 + 400 + 441 = 1225\]

Step 6 ：Divide through by 1225 to get the equation in standard form: \[(x - 4)^2/49 + (y + 3)^2/25 = 1\]

Step 7 ：So, the center of the ellipse is at (h,k) = (4,-3). The lengths of the semi-major and semi-minor axes are \(\sqrt{49} = 7\) and \(\sqrt{25} = 5\) respectively.

Step 8 ：The endpoints of the major axis are at (h ± a, k) = (4 ± 7, -3) = (-3, -3) and (11, -3).

Step 9 ：The endpoints of the minor axis are at (h, k ± b) = (4, -3 ± 5) = (4, -8) and (4, 2).

Step 10 ：\(\boxed{\text{So, the endpoints of the major axis are (-3, -3) and (11, -3), and the endpoints of the minor axis are (4, -8) and (4, 2).}}\)