Answer
Quantities of the form
$kn + kn\log_2n$ for positive integers $k_1 \cdot k_2$ and $n$ arise in the analysis of the merge sort algorithm in computer science. Show that for any positive integer $k$,
$k_1n+k_2n\log_2n$ is $\Theta(n\log_2n)$.
For $n>1$ and for positive integers $k_1,k_2$:
$k_2n\log_2n \leq k_1n+k_2n\log_2n$.
For $n>1$ and for positive integers $k_1,k_2$:
$k_1n \leq k_1n\log_2n$ (by statement 11.4.13)
Add $k_2n\log_2n$ to both sides:
$k_1n + k_2n\log_2n \leq k_1n\log_2n + k_2n\log_2n = (k_1+k_2)\log_2n$.
Because all terms are positive:
$k_2|n\log_2n| \leq |k_1n + k_2n\log_2n| \leq (k_1+k_2)|\log_2n|$.
Hence for $A=k_1, B=(k_1 + k_2), h=1$,
$A|n\log_2n| \leq |k_1n + k_2n\log_2n| \leq B|\log_2n|$ for all $n>h$.
Thus $k_1n+k_2n\log_2n$ is $\Theta(n\log_2n)$
Work Step by Step
Recall the definition of $\Theta$-notation: $f(x)$ is $\Theta(g(x))$ iff there exist positive real numbers A, B, k, such that $A|g(x)| \leq |f(x)| \leq B|g(x)|$ for all $x>k$.
Statement 11.4.13:
For all real numbers $b$ with $b>1$ and for all sufficiently large real numbers $x$:
$x \leq x\log_bx \leq x^2$.