Factoring a trivariate polynomial
I would certainly value some aid with factoring a trivariate polynomial.
The polynomial concerned is
$$p(x,y,z)=a_1 x^7+a_2 x^5y+a_3 x^3y^2+a_4 xy^3+a_5 x^4z+a_6 x^2yz+a_7 y^2z+a_8 xz^2,$$
where the coefficients are integers. I would love to assign values to these coefficients such that $p$ can be factored. As an example, one feasible factorization could be
$$p(x,y,z)=(x^3+b_1 xy+b_2 z)(b_3 x^4+b_4 x^2y+b_5 xz+b_6 y^2).$$
One can expand this factorization and also relate the coefficients of the development with those of the above formula and also hope that the system of formulas can be addressed for integers b's.
Just how can one locate various other feasible factorizations? Exists a very easy means?