For any two statement $p$ and $q,$ the negative of the expression $p\vee \left(~p\wedge q\right)$ is

$~\left(p\vee \left(~p\wedge q\right)\right)$

Using $~\left(a\vee b\right)=~a\wedge ~b, ~\left(a\wedge b\right)=~a\vee ~b \& ~\left(~a\right)=a,$ we get

$=~p\wedge ~\left(~p\wedge q\right)$

$=~p\wedge \left(~\left(~p\right)\vee q\right)$

$=\sim p\wedge \left(p\vee \sim q\right)$

Using distributive law, i.e. $a\wedge \left(b\vee c\right)=\left(a\wedge b\right)\vee \left(a\wedge c\right),$ we get

$=\left(~p\wedge p\right)\vee \left(~p\wedge ~q\right)$

Using $~a\wedge a=C,$ where $C$ represents contradiction, we get

$=C\vee \left(~p\wedge ~q\right)$

Using $C\vee a=a,$ we get

$=~p\wedge ~q.$