Given boolean expression: $(X+Y)(X+\bar Y)+\overline{(X\bar Y)+\bar X}$
$\implies$ $(X.X+X\bar Y+X.Y+Y.\bar Y)+(\overline{(X\bar Y)}*\overline{\bar X})$
$\left [ \because \textrm{Demorgon’s law= $\overline{A+B}=\bar A+\bar B$, Idempotent law= A*A=A}\right]$
$\implies$ $(X+X\bar Y+XY)+(\bar X+\overline{\bar Y})*X)$
taking $X$ as common:
$\implies$ $X(1+\bar Y+Y)+(\bar X+Y)*X$
$\left [ \because A+\bar A=1\textrm{( Complement law)}\right ]$
$\implies$ $X+X*\bar X+X*Y$
$\implies$ $X+XY$
$\implies$ $X(1+Y)$
$\implies$ $X$
Option A is correct.
Ref: Properties of Boolean Algebra