I was solving the equation $3(1+x^2+x^3)^2=(2+x)^4$ for $x$, and after expanding it out, I got $$3x^6+6x^5+2x^4-2x^3-18x^2-32x-13=0\tag{1}$$ which should be solvable because it has a Galois group of order $72$. But since it's a degree six, I have no method for solving this.
I have attempted to factor it into two cubics, but the condition wasn't met, so I can't factor it. This polynomial has (maybe?) irrational roots so the rational root theorem won't work.