MHB Is Every Positive Real Number a Solution to (x+1)(x+2)(x+5)≥36x?

[anemone]

Here is this week's POTW:

-----

Prove that every positive real number satisfies $(x+1)(x+2)(x+5)\ge 36x$.

-----

 

[anemone]

Congratulations to castor28 for his correct solution (Cool) , which you can find below:

Spoiler

We have:
\begin{align*}
(x+1)(x+2)(x+5) - 36x &= x^3 + 8x^2 - 19x + 10\\
&= (x+10)(x-1)^2
\end{align*}
and this is non-negative for all positive $x$ (in fact, for all $x\ge-10$).