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?

$$1+(n+1)x+(n+1)x^2=(1+nx+nx^2)+(x+x^2)<(1+x)^n+x(1+x)<$$
(inductive hypothesis for first inequality)
$$<(1+x)^n+x(1+x)^n=(1+x)^{n+1}$$
($x>0$ for second inequality)
