  In this paper, we introduce a concept of norm-attainment in the projective
symmetric tensor product $\widehat{\otimes}_{\pi,s,N} X$ of a Banach space $X$,
which turns out to be naturally related to the classical norm-attainment of
$N$-homogeneous polynomials on $X$. Due to this relation, we can prove that
there exist symmetric tensors that do not attain their norms, which allows us
to study the problem of when the set of norm-attaining elements in
$\widehat{\otimes}_{\pi,s,N} X$ is dense. We show that the set of all
norm-attaining symmetric tensors is dense in $\widehat{\otimes}_{\pi,s,N} X$
for a large set of Banach spaces as $L_p$-spaces, isometric $L_1$-predual
spaces or Banach spaces with monotone Schauder basis, among others. Next, we
prove that if $X^*$ satisfies the Radon-Nikod\'ym and the approximation
property, then the set of all norm-attaining symmetric tensors in
$\widehat{\otimes}_{\pi,s,N} X^*$ is dense. From these techniques, we can
present new examples of Banach spaces $X$ and $Y$ such that the set of all
norm-attaining tensors in the projective tensor product $X
\widehat{\otimes}_\pi Y$ is dense, answering positively an open question from
the paper by S. Dantas, M. Jung, \'O. Rold\'an and A. Rueda Zoca.
