Computing a mathematical expression using half and full adders
First, realize that multiplying a binary number by a power of 2 can be done by simply shifting the number, and no adders are required. Also, 3B can be written as 2B + B.

So your required calculation becomes

     a1 a0  0  0    (4 A)
           b1 b0    (B)
+       b1 b0  0    (2 B)
----------------
  s4 s3 s2 s1 s0


If you notice that it doesn't require an adder to add 0 to something, you can simplify this to

     a1 a0 b1 b0    (4 A + B)
+       b1 b0  0    (2 B)
----------------
  s4 s3 s2 s1 s0


I believe this can be implemented with 2 half-adders and one full-adder, but I'll leave it to you to work out how and to draw it as a schematic diagram.