Let ${\mathcal P}_n$ be the collection of all polynomials of degree at most $n$ with real coefficients. A subtle Bernstein-type extremal problem is solved by establishing the inequality \[\big\|U_n^{(m)}\big\|_{L_q({\mathbb{R}})} \leq \big(c^{1+1/q}m\big)^{m/2} n^{m/2} \|U_n\|_{L_q({\mathbb{R}})}\] for all $U_n \in \widetilde{G}_n\!$, $q \in (0,\infty]$, and $m=1,2, \ldots$, where $c$ is an absolute constant and $\[ \widetilde{G}_n :=\biggl\{f: f(t) = \sum_{j=1}^N {P_{m_j}(t) e^{-(t-\lambda_j)^2}}, \; \lambda_j \in {\mathbb{R}}, \; P_{m_j}\in\, {\mathcal P}_{m_j}\!, \; \sum_{j=1}^N{(m_j+1)} \leq n \biggr\}. \]$ Some related inequalities and direct and inverse theorems about the approximation by elements of $\widetilde{G}_n$ in $L_q({\mathbb{R}})$ are also discussed.