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If $$a$$, $$b$$ are two real numbers such that $$a + 2b^{1/3} = 5$$, $$a^3 + 8b = 305$$ then
Let $$c = 2b^{1/3}$$, so $$c^3 = 8b$$, $$a + c = 5$$ and $$a^3 + c^3 = 305$$. Using $$a^3 + c^3 = (a + c)^3 - 3ac(a + c)$$, $$305 = 125 - 15ac$$, so $$ac = -12$$, i.e. $$2ab^{1/3} = -12$$ and $$ab^{1/3} = -6$$. Cubing, $$a^3 b = (-6)^3 = -216$$. Hence options a and c are both correct.
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