Your answer is correct.

When you have A < B < C

You have to solve all three inequalities

     A < B,     B < C,    and A < C

and take as the final answer whatever result 
makes all three true at the same time. 

           -2(3 + x) < 4x + 4 < 8x

We do the first and second:

           -2(3 + x) < 4x + 4
             -6 - 2x < 4x + 4
                 -6x < 10
                   x > 
                   x > 
                 
We do the second and third:       

              4x + 4 < 8x
                  4x < 8x - 4
                 -4x < -4
                   x > 1 

We do the first and third:

           -2(3 + x) < 8x
             -6 - 2x < 8x
                -10x < 6
                   x > -6/10
                   x > -3/5

All three of these must be true at the same
time, so therefore x must be greater than 1

So the solution set is

          {x|x > 1}

Graphed on a number line:

 -------------------(============> 
 -5 -4 -3 -2 -1  0  1  2  3  4  5 

Edwin