For taking a series expansion there are two standard ways. One is variable-by-variable.
lpoly = (-2*(132 + 56*x^4 - 382*y + 394*y^2 - 171*y^3 + 26*y^4 + 6*x^3*(-47 + 31*y) +
x^2*(608 - 782*y + 240*y^2) + x*(-602 + 1130*y - 716*y^2 + 153*y^3)))/
((16 + 4*x^2 + 8*x*(-2 + y) - 16*y + 5*y^2)^2*(5*x^2 + (3 - 2*y)^2 + x*(-6 + 4*y))^2);
series1 = Normal[Series[lpoly, {x, 2, 2}, {y, 1, 2}]]
(* Out[29]= -(642/7225) + (-2 + x)^2 (-(17783536/52200625) + (
1232131808 (-1 + y))/887410625 - (56201508164 (-1 + y)^2)/
15085980625) + (-2 + x) (142454/614125 - (8151394 (-1 + y))/
10440125 + (313983542 (-1 + y)^2)/177482125) + (
28698 (-1 + y))/122825 - (900162 (-1 + y)^2)/2088025 *)
The other is to take a series based on total degree, by using an auxiliary variable. We do some forward and backward transformations to center at the origin and then reverse this.
series2 = Normal[Series[lpoly /. {x -> x + 2, y -> y + 1} /. {x -> t*x, y -> t*y},
{t, 0, 2}]] /. t -> 1 /. {x -> x - 2, y -> y - 1}
(* Out[26]= -(642/7225) + (
2 (71227 (-2 + x) + 71745 (-1 + y)))/614125 - (
2 (8891768 (-2 + x)^2 + 20378485 (-2 + x) (-1 + y) +
11252025 (-1 + y)^2))/52200625 *)
