System of equations of polynomial coefficients

Question

I came across a problem recently and I can't seem to figure out why my method isn't working. I have polynomials $$(q_1x^2+q_2x+q_3)(4x+7) + (q_4x^3+q_5x^2+q_6x+q_7)(9) = 1$$ And, when I expand them, I have $$x^3(4q_1+9q_4)+x^2(4q_2+7q_1+9q_5)+x(4q_3+7q_2+9q_6)+(7q_3+9q_7)$$ And then I made a system of equations to try and solve this. $$4q_1+9q_4 = 0$$ $$4q_2+7q_1+9q_5 = 0$$ $$4q_3+7q_2+9q_6= 0$$ $$7q_3+9q_7 = 1$$ If I add all of the systems together, I get $$q_1(4+7)+q_2(4+7)+q_3(4+7)+9q_4+9q_5+9q_6+9q_7 = 1$$ or $$11(q_1 + q_2 + q_3) + 9(q_4+q_5+q_6+q_7) = 1$$ And this is easily solved. Just let $q_1 = 5$ and $q_4 = -6$, and the rest $0$. But when this is plugged back in, $$(5x^2)(4x+7)+(-6x^3)(9) = -34x^3+35x^2 \neq 1$$ I realize that the coefficients sum to $1$, but why is my method of solving the system not valid? If you include the $x^n$ terms, you can divide by them and get the same thing.