On the size of the set A(A+1)

Let $F_p$ be the field of a prime order $p\mathrm{.}$ For a subset $A\subset F_p$ we consider the product set $A(A+1)\mathrm{.}$ This set is an image of $A\times A$ under the polynomial mapping $f(x,y)=xy+x:F_p\times F_p\to F_p\mathrm{.}$ In the present paper we show that if $\mathrm{\mid }A\mathrm{\mid }<p^{1\mathrm{/}2},$ then $$\mathrm{\mid }A(A+1)\mathrm{\mid }\ge \mathrm{\mid }A{\mathrm{\mid }}^{106\mathrm{/}105+o(1)}\mathrm{.}$$ If $\mathrm{\mid }A\mathrm{\mid }>p^{2\mathrm{/}3},$ then we prove that $$\mathrm{\mid }A(A+1)\mathrm{\mid }\gg \sqrt{p\mathrm{\mid }A\mathrm{\mid }}$$ and show that this is the optimal in general settings bound up to the implied constant. We also estimate the cardinality of $A(A+1)$ when $A$ is a subset of real numbers. We show that in this case one has the Elekes type bound $$\mathrm{\mid }A(A+1)\mathrm{\mid }\gg \mathrm{\mid }A{\mathrm{\mid }}^{5\mathrm{/}4}\mathrm{.}$$