Answer
$x-7$
Work Step by Step
Setup:
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.
$\begin{array}{rrr}
{-3 )} &{1}&{-4}&{21}\\
{ } &{ }&{ } &{ }\\
\hline &{1 }&{ } &{ }\end{array}$
Fill the next entries, column by column:
Middle row: k$\times$(previous bottom row entry)
Bottom row: add the two entries above.
Repeat.
$\begin{array}{rrr}
{-3 )} &{1}&{-4}&{21}\\
{ } &{ }&{-3 } &{21 }\\
\hline &{1 }&{ -7} &{0 }\end{array}$
Interpret result: Q(x) is the quotient, R(x) the remainder.
$Q(x)=x-7,\quad R(x)=0$
$\displaystyle \frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$
$\displaystyle \frac{x^{2}-4x-21}{x+3}=x-7$
