Prove hash family is 3-wise independent
Let's assume that $\mathbb{Z}_q$ is the field of size $q$. In order for your family to be 3-wise independent, we need the following condition to hold: for every $x_1,x_2,x_3,y_1,y_2,y_3 \in \mathbb{Z}_q$ where the $x_i$ are all different,
$$
\Pr_s[h_s(x_1) = y_1 \land h_s(x_2) = y_2 \land h_s(x_3) = y_3] = \frac{1}{q^3}.
$$
Since your family consists of $q^3$ functions, this is the same as the following property:


  If $x_1,x_2,x_3 \in \mathbb{Z}_q$ are all different, then for every $y_1,y_2,y_3 \in \mathbb{Z}_q$ there exist unique $s_0,s_1,s_2 \in \mathbb{Z}_q$ such that
  $$ s_0 + s_1 x_i + s_2 x_i^2 = y_i, \quad i \in \{1,2,3\}. $$


This is a property that you can prove using linear algebra.