How to show that inequality is correct

Question

I have that

$$x(1+x)^3\geq (1+x)^2 \geq x(1+x)^2 + (xy)^2 $$

where that $x\in\mathbb{R}$. How does one proceed to show this inequality is true?

Answer 1

The first inequality does not hold for every $x$; indeed, as $(1+x)^2\ge0$, it holds for $x(1+x)\ge1$ or $x=-1$, that is, $x^2+x-1\ge0$, hence for $$ x\le\frac{-1-\sqrt{5}}{2} \qquad\text{or}\qquad x\ge\frac{-1+\sqrt{5}}{2} \qquad\text{or}\qquad x=-1 $$ The second inequality becomes $$ x^3+x^2(1+y^2)-x-1\le0 $$ Since the limit as $x\to\infty$ of the left-hand side is $\infty$, the inequality cannot hold for every $x$ whichever is the value of $y$.