$ | a^3 + ab^2 + a^2 b + b^3 | \leqslant |a + b|^3 $ holds?? Let $a , b \in \mathbb R$. Then $$ | a^3 + a^2 b + ab^2 + b^3 | \leqslant |a + b|^3 $$ holds?

The left hand side is:
$$|a^3+a^2b+ab^2+b^3|=|(a+b)(a^2+b^2)|=|a+b|\cdot(a^2+b^2)$$
The right hand side is:
$$|a+b|^3=|a+b|\cdot(a+b)^2=|a+b|\cdot(a^2+2ab+b^2)$$
So the inequality is true if and only if $RHS-LHS\ge 0$:
$$|a+b|\cdot 2ab\ge 0$$
$$a=-b\quad\text{or}\quad ab\ge 0$$
