How to find a linear combination for a polynomial in multiple variables

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Here's the setup a la Matlab:
>> syms a b c d abcA
>> p1 = expand((a^2+b^2+c^2+d^2+2-2*a*b*c*d)^2)
>> p2 = subs(p1,d^4,d^2*(a^2+b^2+c^2)+2*a*b*c*d)
>> p3 = factor(subs(p2,d^3,d*(a^2+b^2+c^2)+2*a*b*c))
>> q1 = p3 - (d^2+1)*abcA
Here "a" actually refers to sinh(a/2), and the same for b, c, and d. "abcA" refers to (2cosh(a/2)cosh(b/2)cosh(c/2)cos(area/2))^2.
The idea is to completely eliminate d using the following strategy: square and cube q1, producing polynomials q2 and q3, respectively, each of which we can make quadratic in d by repeatedly applying the substitution rule above (d^3 = d*(a^2+b^2+c^2)+2*a*b*c). Then we should be able to find polynomials r1, r2 and r3 in a,b,c and abcA so that r1*q1+r2*q2+r3*q3 is constant in d (ie eliminate the linear and quadratic terms). The desired relation is then 0 = r1*q1+r2*q2+r3*q3.
My question is, how do we find r1, r2, and r3? It turns out that q3 has around 50 second-degree terms in d, which we counted manually, and q2 isn't that much better. We were hoping that there is a simple command or script that might allow us to find the coefficients that will eliminate the linear and quadratic terms.
Thanks so much.

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