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?