How to construct a polynomial which is strictly negative everywhere, except for finitely many roots?

Question

Given a finite sequence of real vectors $x_1, x_2, \dots, x_m \in \mathbb{R}^d$, how do I construct a polynomial $p \colon \mathbb{R}^d \to \mathbb{R}$, such that $p(x) = 0$ if $x \in \{ x_1, \dots, x_m \}$ and $p(x) < 0$ otherwise? Is that even possible? I only know how to construct a polynomial which has roots at $x_1, x_2, \dots, x_m$, but behaves »arbitrarily« elsewhere, e.g. by defining $$ p(x) = \left(\sum_i x_i - x_{1i}\right) \cdot \left(\sum_i x_i - x_{2i}\right) \cdots \left(\sum_i x_i - x_{mi}\right). $$

Answer 1

For example $P(x)=-\prod\limits_{i=1}^m (x-x_i)^2$