Prove Inequality Is True By Induction

Question

$3^n ≥ n^3 +1$ for the integer $n ≥ 4$. I let $n=0$ and the inequality is turns into $1 ≥ 1$ which is true. Then I let $n=n+1$ and the inequality turns into $3^{n+1} ≥ (n+1)^3 +1$. Now from here I get stuck. I tried to expand the right side and I got $3^{n+1} ≥ n^3+3n^2+3n+2$ but still was confused as to how to carry on the next steps from here.

Answer 1

If $3^n\geq n^3+1$ and $n\geq 4$, then: $$ 3^{n+1} = 3\cdot 3^n \geq 3\cdot(n^3+1) \geq (n+1)^3+1 $$ because the last inequality is equivalent to: $$ 2n^3 -3n^2 -3n + 1 = n^2(n-3)+n(n^2-3)+1\geq 0 $$ that is trivial.