, 2016

Expanding the left side of the inequality we get $2a^3 - a^2b - ab^2 + 2b^3$ and factoring it back up again results in
$$\begin{aligned} 2a^3 - a^2b - ab^2 + 2b^3 &= 2(a^3 + b^3) - ab(a + b) = 2(a + b)(a^2 - ab + b^2) - ab(a + b) = \\ &= (a + b)(2a^2 - 3ab + 2b^2). \end{aligned}$$
The second factor is non-negative since $2a^2 - 3ab + 2b^2 = 2(a^2 - \frac{3}{2}ab + b^2) = 2((a - \frac{3}{4}b)^2 + \frac{7}{16}b^2) \ge 0$. So, either $2((a - \frac{3}{4}b)^2 + \frac{7}{16}b^2) = 0$ and $a = b = 0$ or $a + b \ge 0$. The inequality is satisfied by all pairs of reals $a$ and $b$ such that $a + b \ge 0$.