expanding functions with logs to find Big-O estimate

 Please could someone advise on the two questions below. I have to expand the functions
first. Admittedly, I have a weak understanding of logs. Then I try to ascertain a
Big-O estimate. Any help would be appreciated.

1.) (nlogn + n^2)(n^3 +2)

Expanding the above.. Is it correct to say
nlogn(n^3) + nlogn(2) + n^2(n^3) + n^2(2)

==> n^3 (due to the cancellation laws) +2nlogn + n^5 +2n^2
==> therefore a Big-O estimate would be n^5 ??

2.)  nlog(n^2 +1) + n^2logn
expanding the above ..is it correct to say
2nlogn + nlogn + n^2logn
As n<n^2logn<n^2,
the Big - O estimate is (n^2)?

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On the first problem, the dominant term is $n^5$, so it's $O(n^5)$. On the second one, your dominant termis $n^2\log(n)$ so it is $O(n^2\log(n))$.