What kind of inequality is this? What kind of inequality is this?
$|z^2 + 1|$ is greater than or equal to $|z|^2 - 1$
why $-1$? , and why not $-100\,000$?
does $z$ have to be a complex variable?
correction : sorry it was not $|z^2|$, it was $|z|^2$

In the reals,
$$z^2+1\ge z^2$$ holds (and the modulus is not even necessary).
But in the complex, a counter-example is
$$|i^2+1|=|i^2|-1=0.$$

Let $z^2=a+ib$ and let us look for the minimum of
$$\sqrt{(a+1)^2+b^2}-\sqrt{a^2+b^2}.$$
We cancel the gradient,
$$\begin{cases}\dfrac{a+1}{\sqrt{(a+1)^2+b^2}}=\dfrac{a}{\sqrt{a^2+b^2}},
\\\dfrac{b}{\sqrt{(a+1)^2+b^2}}=\dfrac{b}{\sqrt{a^2+b^2}}\end{cases}$$
and this implies $b=0$, and $z^2$ is real. The function reduces to
$$|a+1|-|a|$$ which has the minimum $-1$. Hence
$$|z^2+1|\ge|z^2|-1$$ is tight.
