$\sigma_1\sigma_2 =\Sigma_{i<j}{~~(z_i^2z_j+z_iz_j^2})+3\Sigma_{i<j<k}{~~z_iz_jz_k}$

Question

Let $P\in\mathbb{R}[X]$ of degree $n$ and $z_1,...,z_n $ the complex roots of $P$. We consider that $\sigma_1(z_1,...,z_n)=\Sigma_{i}z_i$ and $\sigma_2(z_1,...,z_n)=\Sigma_{i<j}z_iz_j$. I read in a book that $$ \sigma_1\sigma_2 =\Sigma_{i<j}{~~(z_i^2z_j+z_iz_j^2})+3\Sigma_{i<j<k}{~~z_iz_jz_k} $$ But I can not see that. Help me please