Algebra

$a^2 + b^2 = 6 + 2ab\\ a^2 + 2ab + b^2 = 6 + 4ab\\ (a + b)^2 = 6 + 4ab$

But a + b = 4, so

$4^2 = 6 + 4 ab\\ ab = \dfrac 52$

Then consequently $a^3 + b^3 = (a + b)^3 - 3ab(a + b) = 4^3 -3\cdot \dfrac52 \cdot 4 = 34$.