Answer
$2a+4+\dfrac{5}{a+2}$
Work Step by Step
Setup: (the variable is $a$ instead of $x$)
Dividing with $x-k$, place k as the top left entry.
List coefficients of the numerator (0 for missing powers) in the first row.
Copy the leading coefficient to the bottom row, same column.
$\begin{array}{rrr}
{-2 )} &{2}&{8}&{13}\\
{ } &{ }&{ } &{ }\\
\hline &{2 }&{ } &{ }\end{array}$
Fill the next entries, column by column:
Middle row: k$\times$(previous bottom row entry)
Bottom row: add the two entries above.
Repeat.
$\begin{array}{rrr}
{-2 )} &{2}&{8}&{13}\\
{ } &{ }&{-4 } &{ -8 }\\
\hline &{2 }&{ 4} &{5 }\end{array}$
Interpret result:
Q(x) is the quotient, R(x) the remainder. $\displaystyle \frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$
$Q(a)=2a+4,\quad R(a)=5$
$\displaystyle \frac{2a^{2}+8a+13}{a+2}=2a+4+\frac{5}{a+2}$
