  Let $r\in\mathbb{N}$ and $\mathbb{N}=\bigcup_{i=1}^{r}C_{i}$. Do there exist
$x,y\in\mathbb{N}$ and $r\in\left\{1,2,\ldots,r\right\}$ such that
$\left\{x,y,xy,x+y\right\}\subseteq C_{i}$? This is still an unanswered
question asked by N. Hindman. Joel Moreira in [J. Moreira, Monochromatic sums
and products in N, Annals of Mathematics 185 (2017) 1069-1090] established a
partial answer to this question and proved that for infinitely many
$x,y\in\mathbb{N}$, $\left\{x,xy,x+y\right\}\subseteq C_{i}$ for some
$i\in\left\{1,2,\ldots,r\right\}$, which is called Moreira's Theorem. Recently,
H. Hindman and D. Strauss established a refinement of Moreira's Theorem and
proved that for infinitely many $y$,
$\left\{x\in\mathbb{N}:\left\{x,xy,x+y\right\}\subseteq C_{i}\right\}$ is a
piecewise syndetic set. In this article, we will prove infinitely many $y\in
FP\left(\mathbb{P}-1\right)$ such that
$\left\{x\in\mathbb{N}:\left\{xy,x+f(y):f\in F\right\}\subseteq C_{i}\right\}$
is piecewise syndetic, where $F$ is a finite subset of
$x\mathbb{Z}\left[x\right]$. $\mathbb{P}$ is the set of prime numbers in
$\mathbb{N}$ and $FP\left(\mathbb{P}-1\right)$ is the set of all finite
products of $\mathbb{P}-1$.
