To solve the equation \( 3(x + 4) = 3x + 4 \), we can start by simplifying both sides.
First, expand the left side:
\[ 3(x + 4) = 3x + 12 \]
Now, rewrite the equation with the expanded left side:
\[ 3x + 12 = 3x + 4 \]
Next, we can subtract \( 3x \) from both sides:
\[ 12 = 4 \]
This statement, \( 12 = 4 \), is false. Since we arrived at a contradiction, it indicates that there are no values of \( x \) that can satisfy the original equation.
Thus, the number of solutions to the equation \( 3(x + 4) = 3x + 4 \) is
\[ \boxed{0}. \]