Let domain of $x = \{x_1, x_2\}$
Option A:
$[\forall x (P(x) \lor W) \equiv \forall x P(x) \lor W]$
$LHS: \forall x (P(x) \lor W) =(P(x_1) + W) * (P(x_2) + W) =P(x_1)P(x_2) + W$
$ RHS: \forall x P(x) \lor W = P(x_1)*P(x_2) + W$
$\therefore LHS \equiv RHS$
Option B:
$[\exists x (P(x) \land W) \equiv \exists x P(x) \land W ]$
$LHS: \exists x (P(x) \land W) = (P(x_1).W)+(P(x_2).W) = \{P(x_1)+P(x_2)\}*W$
$RHS: \exists x P(x) \land W = \{P(x_1) + P(x_2)\} * W$
$\therefore LHS \equiv RHS$
Option C:
$[\forall x (P(x) \rightarrow W) \equiv \forall x P(x) \rightarrow W]$
$LHS: \forall x (P(x) \rightarrow W) = \{(P(x_1) \rightarrow W)*(P(x_2) \rightarrow W)\} = (P’(x_1) + W)(P’(x_2) + W) = P’(x_1)P’(x_2) + W$
$RHS: \forall x P(x) \rightarrow W = (P(x_1)P(x_2)) \rightarrow W = (P(x_1)P(x_2)’ + W = P’(x_1) + P’(x_2) + W$
$\therefore LHS \not \equiv RHS$
Option D:
$[\exists x (P(x) \rightarrow W) \equiv \forall x P(x) \rightarrow W]$
$LHS: \exists x (P(x) \rightarrow W) = \{(P(x_1) \rightarrow W) + (P(x_2) \rightarrow W)\} = P’(x_1) + W + P’(x_2) + W = P’(x_1) + P’(x_2) + W$
$RHS: \forall x P(x) \rightarrow W = (P(x_1)*P(x_2)) \rightarrow W = (P(x_1)P(x_2))’ + W = P’(x_1) + P’(x_2) + W$
$\therefore LHS \equiv RHS$