Here's a formal induction proof of $T(n)=O(n^3)$ in LaTeX:

\begin{proof}
We want to show that there exist positive constants $c$ and $n_0$ such that for all $n \geq n_0$, we have $T(n) \leq c \cdot n^3$.

We can choose $c = 15 + 1 + 4/n_0^3$ and $n_0 = 2$. Then, for all $n \geq n_0$,

\begin{align*}
T(n) &= 15n^3 + n^2 + 4 \
&\leq 15n^3 + n^3 + n^3 \
&= (15+1+1)n^3 \
&\leq 17n^3 \
&\leq c \cdot n^3.
\end{align*}

Therefore, $T(n) = O(n^3)$.
\end{proof}
spam fails_task lang_mismatch pii not_appropriate hate_speech sexual_content quality toxicity humor helpfulness creativity violence