Did I do this correctly? And what conic does the equation represent?

(2/5x^2)-2y+4x+3y^2-4=0

(2/5x^2+ 4x +__) + (3y^2 -2y+__)= 4+__ + __

2/5(x^2 + 10x +25) + 3(y^2 - 2/3y+ 1/9)= 4 + 10 + 1/3

2/5(x+5)^2 + 3(y-1/3)^2 =43/3

[6(x+5)^2]/215 + [9(y-1/3)^2]/43 = 1

Conic: I think ellipse

If I did this incorrectly please correct my mistake thank you!