Question

x²+4y²+2x-8y+1=0

Answer 1

What about it,
looks like you have an ellipse.

I will assume you want to find the properties after completing the square

x^2 + 2x + ... + 4y^2 - 8y + .... = -1 + .....
x^2 + 2x + 1 + 4(y^2 - 2y + 1) = -1 + 1 + 4
(x + 1)^2 + 4(y - 1)^2 = 4

divide each term by 4
(x+1)^2 / 4 + (y - 1)^2 = 1

draw any conclusions from this