I have the function $f(x)={{x^3+9}\over{x^2}}$ and I'm trying to prove that it is injective in $(-\infty, 0)$ through the definition.

**What I've done:**

$
f(a)=f(b)\Leftrightarrow
{{a^3+9}\over{a^2}}={{b^3+9}\over{b^2}}\Leftrightarrow
b^2(a^3+9)=a^2(b^3+9)\Leftrightarrow
a^3b^2+9b^2=a^2b^3+9a^2\Leftrightarrow\\
a^3b^2-a^2b^3=9a^2-9b^2\Leftrightarrow
a^2b^2(a-b)=9(a-b)(a+b)\Leftrightarrow
a^2b^2=9(a+b)
$

From this point on, I'm pretty much doing circles without ever reaching $f(a)=f(b)\Leftrightarrow a=b$.

**Question:**

How can I prove that $f(x)$ is an injective function? Any tip that will help me get unstuck from where I am and points to me the right direction will be greatly appreciated.