Inequality proof of integers

Question

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?

Answer 1

$$1+(n+1)x+(n+1)x^2=(1+nx+nx^2)+(x+x^2)<(1+x)^n+x(1+x)<$$ (inductive hypothesis for first inequality)

$$<(1+x)^n+x(1+x)^n=(1+x)^{n+1}$$

($x>0$ for second inequality)

Answer 2

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