Prove that: $$(n+3)^2 \le 2^{n+3},\quad n\in\mathbb{N}$$
Please show me how to prove this inequality using a method other than mathematical induction. I was solving some questions based on the principle of mathematical induction and after solving nearly $20$ questions, I noticed that there is always an alternate proof for a statement which I proved by using principle of mathematical induction. So, I was trying to prove the statement given in the title by an alternate method. I found that the inequality can be easily verified from the graph, but I wanted to know if it can be proved in any algebraic way without using graphs.
Look at $(x+3)\ln 2-2\ln (x+3)$. It is increasing on $[1,\infty)$. Now can you finish?