Find the square values of a polynomial

Hello there,

I am trying to find all integers for which the polynomial

P(x) = 4x^6 + 36x^5 + 136x^4 + 276x^3 + 328x^2 + 228x + 81

is a perfect square. I checked this for -1000000 <= x <= 1000000 and I
got only x=-4,-3,-2,-1,0,1. However, I could not
prove that these are the only values nor find other values. Could
someone help with this.


Thanks.

.