## Write a polynomial in terms of elementary symmetric polynomials

Let $p(x) = \prod_{i,j = 1}^2 (x - a_ib_j)$. Verify that the coefficients of $p(x)$ can be written as polynomials in the elementary symmetric polynomials in $\alpha_i$ and $\beta_j$.

Let $\sigma_1 = a_1+a_2$, $\sigma_2 = a_1a_2$, $\tau_1 = b_1+b_2$, and $\tau_2 = b_1b_2$. Evidently, $p(x) = x^4 - \sigma_1\tau_1x^3 + (\sigma_1^2\tau_2 + \sigma_2\tau_1^2 - 2\sigma_2\tau_2)x^2 - \tau_1\tau_2\sigma_1\sigma_2 x + \sigma_2^2 \tau_2^2$.