Determine the type of conic to which the equation belongs: 12y^2-72y+x+78=0

Solution variant #1.

25x² + 9y² + 72y - 81 = 0, 25x² + 9y² + 72y - 81 + 81 = 0 + 81 , 25x² + 9y² + 72y = 81, 25x² + 9(y² + 8y + 16) = 81 + 9(16) , 25x² + 9(y² + 4y + 4y + 16) = 81 + 144 , 25x² + 9(y(y) + 4(y) + 4(y) + 4(4)) = 225 , 25x² + 9(y(y + 4) + 4(y + 4)) = 225 , 25x² + 9(y + 4)(y + 4) = 225 , 25x² + 9(y + 4)² = 225
The conic section is an ellipse."