Sure, here's a formal induction proof to show that $T(n)=O(n^3)$ using the Big O notation:

Let $c = 15$ and $n_0 = 1$. Then, for all $n \ge n_0$ we have:

$$T(n) = 15n^3 + n^2 + 4 \le 15n^3 + n^3 + n^3 = 31n^3.$$

Thus, we have $T(n) \le 31n^3$ for all $n \ge n_0$, and it follows that $T(n) = O(n^3)$.

Here's how the above proof would look in LaTeX:
Let $c = 15$ and $n_0 = 1$. Then, for all $n \ge n_0$, we have:

$$T(n) = 15n^3 + n^2 + 4 \le 15n^3 + n^3 + n^3 = 31n^3.$$

Thus, we have $T(n) \le 31n^3$ for all $n \ge n_0$, and it follows that $T(n) = O(n^3)$.
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