Let's rewrite all terms with the same base, and let's make 2 the common base.
Given: 4^a x 4^b = 2(8^3)
Rewrite 4 and 8 as powers of 2: (2^2)^a x (2^2)^b = 2((2^3)^3)
Simplify: 2^(2a) x 2^(2b) = (2)(2^9)
Simplify again: 2^(2a + 2b) = 2^10
Since the bases are now equal, we can conclude that 2a + 2b = 10
Divide both sides by 2 to get: a + b = 5

Answer: B

In this case, I do not see the advantage of distributing the 2 into (a+b) since the question is asking what that quantity is. We can consider it a single variable in this case, right?

After making the same base to equate the exponents, ended with

2(a+b)=10

.

Same answer in the end, but eliminating a potential place for error.

2(a+b)=10
a+b=5