How to solve a conic math problem

Identify the type of conic, if any. If a conic, state it's vertex.

3x^2 + 12x + 4y^2 - 40y + 100 = 0

Please tell me how you did it! Thanks!

3x^2 + 12x + 4y^2 - 40y + 100 = 0
3 * (x^2 + 4x) + 4 * (y^2 - 10y) = -100
3 * (x^2 + 4x + 4 - 4) + 4 * (y^2 - 10y + 25 - 25) = -100
3 * (x^2 + 4x + 4) - 12 + 4 * (y^2 - 10y + 25) - 100 = -100
3 * (x + 2)^2 + 4 * (y - 5)^2 = 12
(x + 2)^2 / 4 + (y - 5)^2 / 3 = 0

It's an ellipse.

The vertices are at (-4 , 5) , (0 , 5) , (-2 , 5 + sqrt(3)) , (-2 , 5 - sqrt(3))