Solution:

The inequality is homogeneous, so we can take $a = 1$ and put $b = 1 + x$, $c = 1 + y$, where $x, y \geq 0$. Then after some reduction the inequality is equivalent to $x^{3} + y^{3} + x^{2} + y^{2} - x^{2} - y - xy^{2} - xy \geq 0$, or (after factorising $x^{3} + y^{3}$) to $(x + y + 1)(x - y)^{2} + xy \geq 0$, which is obviously true.