  Under certain hypotheses on the Banach space $X$, we show that the set of
$N$-homogeneous polynomials from $X$ to any dual space, whose Aron-Berner
extensions are norm attaining, is dense in the space of all continuous
$N$-homogeneous polynomials. To this end we prove an integral formula for the
duality between tensor products and polynomials. We also exhibit examples of
Lorentz sequence spaces for which there is no polynomial Bishop-Phelps theorem,
but our results apply. Finally we address quantitative versions, in the sense
of Bollob\'as, of these results.
