# How do we obtain the coefficients of quadratic polynomials in a product term?

Assume we are given quadratic polynomials $$p_i(x) = 1+a_i \cdot x + b_i \cdot x^2,$$ for $$i = 1,\ldots,n$$ with $$a_i,b_i,x$$ being real numbers.

If we consider now the product $$P(x) = \prod_{i=1}^\infty p_i(x)$$ and do know only the coefficients of $$P$$, i.e., $$P(x) = 1+c_1 \cdot x + \ldots + c_{2n} x^{2n}$$, can we then gain $$a_i, b_i$$, solely by knowing the $$c_i$$?

I think of it as follows: The $$a_i$$ and $$b_i$$ are related to the $$2n$$ values of $$c_i$$ by algebraic equations like $$c_1 = a_1 + a_2 \ldots + a_n$$ and $$c_{2n} = b_1 \cdot b_2 \ldots \cdot b_n,$$ but how can we solve this equation system?