Since Mathematica 10 there's been ImplicitRegion. We define that region this way:

IR = ImplicitRegion[-9 <= 6 x + 5 y <= 9 && -7 <= 3 x + 6 y <= 7, {x, y}]; 

then we can calculate this integral directly:

Integrate[(18 x^2 + 51 x*y + 30 y^2)^2, {x, y} ∈ IR] 

5292 

Without the newest version of the system then this integral can be calculated in the following way:

18 x^2 + 51 x*y + 30 y^2 // Factor 

3 (x + 2 y) (6 x + 5 y) 

We change linearly variables: w == 3x + 6y; z == 6x + 5y;. The Jacobian of this transformation we calculate directly:

Det @ D[{3 x + 6 y, 6 x + 5 y}, {{x, y}}] 

-21 

and the integral can be factorized, thus :

int = 1/21 Integrate[w^2 z^2, {w, -9, 9}, {z, -7, 7}] 

 5292 

Since Mathematica 10 there's been ImplicitRegion. We define that region this way:

IR = ImplicitRegion[-9 <= 6 x + 5 y <= 9 && -7 <= 3 x + 6 y <= 7, {x, y}]; 

then we can calculate this integral directly:

Integrate[(18 x^2 + 51 x*y + 30 y^2)^2, {x, y} ∈ IR] 

5292 

This integral can be calculated without the newest version of the system, to enlighten its structure we proceed in the following way:

18 x^2 + 51 x*y + 30 y^2 // Factor 

3 (x + 2 y) (6 x + 5 y) 

We change linearly variables: w == 3x + 6y; z == 6x + 5y;. The Jacobian of this transformation we calculate directly:

Det @ D[{3 x + 6 y, 6 x + 5 y}, {{x, y}}] 

-21 

We take the absolute value of its inverse and the integral can be factorized, thus :

int = 1/21 Integrate[w^2 z^2, {w, -9, 9}, {z, -7, 7}] 

 5292 

For completness we provide another two straightforward ways which work also in earlier versions of the system:

Integrate[ Boole[-9 <= 6 x + 5 y <= 9 && -7 <= 3 x + 6 y <= 7] (18 x^2 + 51 x y + 30 y^2)^2, {x, -∞, ∞}, {y, -∞, ∞}] Integrate[ HeavisideTheta[6 x + 5 y + 9] HeavisideTheta[9 - 6 x - 5 y] HeavisideTheta[3 x + 6 y + 7] HeavisideTheta[7 - 3 x - 6 y] (18 x^2 + 51 x y + 30 y^2)^2, {x, -∞, ∞}, {y, -∞, ∞}]