Magma says it's irreducible:
Code:
P<x,y> := PolynomialRing(IntegerRing(), 2);
Factorization(216*x^2*y^2+150*x^2+6*y^2+72*x*y^2+360*x^2*y+48*x*y+40*x+8*y+7);
[
<216*x^2*y^2 + 360*x^2*y + 150*x^2 + 72*x*y^2 + 48*x*y + 40*x + 6*y^2 + 8*y
+ 7, 1>
]
Total time: 0.140 seconds, Total memory usage: 7.28MB