How to prove Big Theta on polynomial function?

Question

Im working on the Big-O notation and struggle to understand whether I have done enough to prove the following:

$5n^3+4n^2+4 \in \Theta (n^3)$

So based on the definition of $Θ(g(n)):$

Step $1$: $0 ≤ c_1 n^3 \leq 5n^3+4n^2+4 \leq c_2 n^3$

Divide the inequality by the largest order n-term

Step $2$: $0 ≤ c_1 \leq 5+(4/n+4/n^3) \leq c_2$

Find constant $c_2$ that will satisfy:

Step $3$: $0 \leq 5+(4/n+4/n^3) \leq c_2$

For $n=1$, $0 \leq 5+(4/1+4/1^3)=13$

For $n=2$, $0 \leq 5+(4/2+4/2^3)=7,5$

For $n=3$, $0 \leq 5+(4/3+4/3^3)=6,7$

For $n=4$, $0 \leq 5+(4/4+4/4^3)=6,06$

Since $c_2$ approaches $5$ when $n \to \infty$, I pick 5 for $c_2$ as it satisfies the equation in Step $3$

Step $4$: Is to find $c_1$

Since $c_1$ can be $\leq 5$

I can say that $5n^3+4n^2+4 \in \Theta (n^3)$ for $c_1$ $\leq 5$ and $c_2 \geq 5$. However not sure how to find $n_0$ in that regard.

Appreciate any insights!

Answer 1

Hint:   $1 \le n^2 \le n^3$ for $n \ge 1\,$, so $5n^3 \;\le\; 5n^3+4n^2+4 \;\le\; 5n^3+4n^3+4n^3 = 13 n^3\,$.

Answer 2

this is a good solution: https://walkccc.github.io/CLRS/Chap03/Problems/3-1/ note: [d * 1/d > 1/d+1/d^2+....+1/d^d , as there are d terms at right hand side, and all of them are less than 1/d]