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We know $ a + b = 4 $, and
a^2 + b^2 = (a + b)^2 - 2ab = 16 - 2ab = 10 \Rightarrow 2ab = 6 \Rightarrow ab = 3
a^3 + b^3 = 4^3 - 3 \cdot 3 \cdot 4 = 64 - 36 = \boxed{28}

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