If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 4,$ then find
\[\begin{vmatrix} a & 7a + 3b \\ c & 7c  +3d \end{vmatrix}.\]
Since $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 4,$ $ad - bc = 4.$  Then
\[\begin{vmatrix} a & 7a + 3b \\ c & 7c  +3d \end{vmatrix} = a(7c + 3d) - (7a + 3b)c = 3ad - 3bc = 3(ad - bc) = \boxed{12}.\]
Since $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 4,$ $ad - bc = 4.$  Then
\[\begin{vmatrix} a & 7a + 3b \\ c & 7c  +3d \end{vmatrix} = a(7c + 3d) - (7a + 3b)c = 3ad - 3bc = 3(ad - bc) = \boxed{12}.\]
The final answer is $$\boxed{12}$$. I hope it is correct.