Since $a(x+b)=3 x+12$ for all $x$, then $a x+a b=3 x+12$ for all $x$. Since the equation is true for all $x$, then the coefficients on the left side must match the coefficients on the right side. Therefore, $a=3$ and $a b=12$, which gives $3 b=12$ or $b=4$. Finally, $a+b=3+4=7$.