<s>[INST] Show that $x^3 - 2 x^2 +\log(1+x)(x(3x+4) -2(1+x)^2 \log(1+x))$ is positive I want to show that (the following just gets rid off large brackets)
$$x^3 - 2 x^2 +x(3x+4)\log(1+x)-2(1+x)^2 \log^2(1+x)>0, \ \ \mbox{for}\ \ x\in(0,\infty).$$
My attempt: Transfer all negative terms to the other side of the inequality. Evaluate the both sides at different limits of $x$, and see that both sides converge to $0$ for $x \to 0$ and increase infinitely for $x \to \infty$.  Take the derivative of both sides with respect to $x$ to demonstrate that the left side increases faster than the right side.  Burn out at that point because showing that the last point holds I need to engage into a similar task as proving the original inequality.
 [/INST] Solve with respect to $\log(1+x)$ to obtain
\begin{equation}
 \frac{3x^2 + 4x - x^2\sqrt{8x+9}}{4(1+2x+x^2)}<\log(1+x)<\frac{3x^2 + 4x + x^2\sqrt{8x+9}}{4(1+2x+x^2)}, \ \ \mbox{for}\ \ x>0. \tag{*}
\end{equation}
Start with the first inequality. Observe that both sides of the inequality go to zero when $x\to 0$.  Also it is easy to check the LHS (containing the fraction term) is increasing starting from zero and then is decreasing in $x$, while the RHS (containing $\log$) is increasing in $x$ and goes to infinity when $x\to \infty$.  Therefore, the inequality holds if the RHS is increasing faster than the LHS, or formally (taking derivatives, respectively):
$$\frac{-10 x^2 - 2 x^3 + 2\sqrt{9 + 8 x} + x (-9 + \sqrt{9 + 8 x})}{2(1+x)^3\sqrt{9+8x}} < \frac{1}{1+x}.$$
Proving the last inequality holds should not be a problem.  Hence, the first inequality of $(*)$ is true.
Do similar steps to show that the second inequality in $(*)$ holds.
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