$3.5$n$2+3$nlog$2$n$+2$n$+5$ is O$(?).$

$(1)$ What should be the question mark? $(5$ points$)\backslash [$ O$($n$^2)\backslash ]$

$(2)$ Prove your answer in the question mark using the two methods that we learned during classes. As a reminder, first method is to prove by definition $(5$ points$)$; the second method is to prove by limits in mathematics $(5$ points$).$

$($Total points in question $3$ are $15$ points$)$