Inequality proof of integers My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook.
Page 36. Exercise 7. Let $n_1$ be the smallest positive integer $n$ for witch the inequality $(1+x)^n>1+nx+nx^2$ is true for all $x>0$. Compute $n_1$, and prove that the inequality is true for all integers $n\ge n_1$.
The attempt at a solution: I solved first question asked, which was to find the value of $n_1$, it is equal to $3$, for the second part, I am assuming that I have to prove the inequality by induction, since the chapter is about induction, here's my attempt: $$(1+x)^{n+1}=(1+x)^n(1+x)>(1+nx+nx^2)(1+x)=nx^2(x+2)+(n+1)x+1$$Which gets me nowhere, what am I doing wrong?

With $n,x>0$,
$$
nx^2(x+2)-(1+n)x^2=nx^2(x+1)-x^2=x^2(n(x+1)-1)>0\\
\implies nx^2(x+2)>(1+n)x^2.
$$
