The dominant term is $n^3$, so the runtime is $O(n^3)$.

Let's simplify the expression; this requires certain level of mathematical literacy. Most of what is done below is based on the laws of logarithms.

$$ = 3\times 2\lg n + \lg^2 n + (n^2 + 2n + 1)(\lg 4 + \lg n) $$

$$ = 6\lg n + \lg^2 n + (n^2+2n+1)(2+\lg n) $$

$$ = 6\lg n + \lg^2 n + 2n^2 + 4n + 2 + n^2\lg n + 2n\lg n + 2\lg n $$

$$ = n^2\lg n + 2n^2 + 2n\lg n + 4n + \lg^2 n + 8\lg n + 2 $$

The dominant term is $n^2\lg n$, so the runtime is $O(n^2\lg n)$.

Note, $\lg n$ is $\log_2 n$ and $n^2\lg n$ is larger than $2n^2$ for all $n > 4$.

You might be wondering how I know which term has the highest order (fastest rate of growth). This, again, requires certain level of mathematical literacy. However, there are a dozen of functions that show up in most practical cases and we will survey them in a later lesson.