Integrate[
Boole[{ -2 <= 2 x + 7 y <= 2 && -2 <= 3 x + 6 y <= 2}](6 x^2 +
33 x*y + 42 y^2)^4, {x, -∞, ∞}, {y, -∞, ∞}]
4096/225
Integrate[(6 x^2 + 33 x*y + 42 y^2)^4, {x, y} ∈
ImplicitRegion[{ -2 <= 2 x + 7 y <= 2 && -2 <= 3 x + 6 y <= 2}, {x,
y}]]
4096/225
(* change of variables *)
sol = Solve[{u == 2 x + 7 y, v == 3 x + 6 y}, {x, y}][[1]];
(* {x -> 1/9 (-6 u + 7 v), y -> 1/9 (3 u - 2 v)} *)
expr = (6 x^2 + 33 x*y + 42 y^2)^4 /. sol // Simplify;
(* u^4 v^4 *)
jacobian = Grad[{x, y} /. sol, {u, v}] // Det // Abs
(* 1/9 *)
Integrate[expr*jacobian, {u, -2, 2}, {v, -2, 2}]
4096/225
