find the coordinate of the center and identify the conic:
5y^2 - 2x^2 - 10y - 12x = 23
Complete the squares of each squared term:
5y^2 - 2x^2 - 10y - 12x = 23
5(y-1)^2-5 -2(x+3)^2+18 = 23
5(y-1)^2 -2(x+3)^2 = 10
((y-1)/sqrt(2))^2 -((x+3)/sqrt(5))^2=1
From the above, the centre is located at (-3,1) and it is a hyperbola expressed in the form ((x/a)^2-(y/b)^2=1).
Check my algebra.