<s>[INST] how to prove by induction the $ (1+x)^{n}>1+nx+nx^2$ Prove by induction the formula 
$ (1+x)^{n}>1+nx+nx^2$ for $x>0$ real number and $n\ge 3$
my try : multiply both sides by $(1+x)$
gives $ (1+x)^{n+1}>1+(n+1)x+(2n+nx)x^2$
have I done something wrong or what I do next?
$(2n+nx)$ must turn into $(n+1)$
 [/INST] You are almost there. Note that $2n+x \geq 2n \geq n+1$.
Hence, the induction step becomes:
We have
$$(1+x)^n > 1 + nx + nx^2$$
Multiplying by $1+x$, we obtain
\begin{align}
(1+x)^{n+1} & = (1+x)^n (1+x)\\
& > (1+nx+nx^2)(1+x)\\
& = 1+ nx + nx^2 + x + nx^2 + nx^3\\
& = 1+(n+1)x + (n+n)x^2 + nx^3\\
& \geq 1+(n+1)x + (n+n)x^2 & (\because nx^3 \geq 0)\\
& \geq 1+(n+1)x + (n+1)x^2 & (\because n \geq 1)
\end{align}
which gives us the induction step.
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