System

First, look at the first equation. Squaring both sides, we have

$a^2 + 2ab + b^2 = 16$

Now, we have two equations to work with. We have

$a^2 + b^2 = 16 - 2ab \\ a^2 + b^2 = 6 + 2ab$

Subtracting the second equation from the first equation, we get

$0 = -10 + 4ab \\ 10 = 4ab \\ ab = 10 / 4 = 5/2$

Now, let's acknowledge something about a^3+b^3. Note that

$a^3 + b^3 = (a + b) ( a^2 + b^2 - ab)$

Wait! we already have all the terms needed to solve the problem!

We have

$\\ (4) ( 6 + 2ab - ab) = \\ (4) ( 6 - ab) = \\ (4) ( 6 - 5/2) = \\ (4) ( 7/2) = \\ 14$

So 14 is our answer. 

Thanks! :)