To simplify the expression \((4 + 7w)(2)\), distribute the \(2\) to both terms inside the parentheses:
\[ (4 + 7w)(2) = 4 \cdot 2 + 7w \cdot 2 \]
Calculating each part:
\[ 4 \cdot 2 = 8 \] \[ 7w \cdot 2 = 14w \]
Now combine these results:
\[ 8 + 14w \]
Thus, the simplified expression is:
\[ \boxed{8 + 14w} \]