Cases in which a certain inequality holds true.

Question

I previously asked this question on this forum, and have been demonstrated counterexamples to the claim that $|a| > |b|$ implies $\big|\frac{b+b^{2}}{a+a^{2}}\big| < 1$, which I had previously thought to be the case. I am now wondering, for which values of $a$ and $b$ is our claim true, and for which values is our claim false. For example, our claim is obviously true for $a \geq 1, b \geq 1$, and is obviously true for $a \leq 1, b \leq 1$. An example where our claim is false would be for $a = -1.01, b = 1$. Knowing precisely where our claim holds true is of great interest to me for a scientific problem I am looking at, so a formal argument would be greatly appreciated. Thank you.