Answer
$x^{2}+2x+9$
Work Step by Step
When dividing a polynomial $f(x)$ with $(x-k)$
We set up syntehetic division $\quad \text{divisor } )\overline{\text{ dividend }}$,
by placing $k$ in place of the divisor,
and listing ALL coefficients of $f(x)$ (including the zeros ), starting from highest power of x.
$ \begin{array}{lllllll}
& & -- & -- & -- & -- & \\
-1 & ) & 1 & 3 & 11 & 9 & \\
& & & -1 & & & \\
& & -- & -- & -- & -- & \\
& & 1 & & & & \\
& & & & & &
\end{array}$
We are ready.
Bring down the leading coefficient, $1$
Multiply $(-1)(1)=-1$ and place it in the next free slot of the middle row.
$ \begin{array}{lllllll}
& & -- & -- & -- & -- & \\
-1 & ) & 1 & 3 & 11 & 9 & \\
& & & -1 & & & \\
& & \downarrow--\nearrow & -- & -- & -- & \\
& & 1 & & & & \\
& & & & & &
\end{array}$
Add $3 $and $-1$ and place $2$ in the bottom row.
Multiply $(-1)(2)=-2$ and place it in the next free slot of the middle row.
$ \begin{array}{lllllll}
& & -- & -- & -- & -- & \\
-1 & ) & 1 & 3 & 11 & 9 & \\
& & & -1 & -2 & & \\
& & -- & --\nearrow & -- & -- & \\
& & 1 & 2 & & & \\
& & & & & &
\end{array}$
Place $11+(-2) =9$ in the bottom row.
Multiply $(-1)(9)=-9$ and place it in the next free slot of the middle row.
$ \begin{array}{lllllll}
& & -- & -- & -- & -- & \\
-1 & ) & 1 & 3 & 11 & 9 & \\
& & & -1 & -2 & -9 & \\
& & -- & -- & --\nearrow & -- & \\
& & 1 & 2 & 9 & & \\
& & & & & &
\end{array}$
Add $9+(-9)=0$ and place it in the bottom row.
$ \begin{array}{lllllll}
& & -- & -- & -- & -- & \\
-1 & ) & 1 & 3 & 11 & 9 & \\
& & & -1 & -2 & -9 & \\
& & -- & -- & -- & -- & \\
& & 1 & 2 & 9 & 0 & \\
& & & & & &
\end{array}$
Interpret the result:
The last number of the bottom row represents the remainder, $0.$
The rest of the botom row holds coefficients of the quotient, which has a degree of one less than $f(x):\quad q(x)=x^{2}+2x+9$
So, $\displaystyle \quad \frac{f(x)}{x-k}=q(x)+\frac{r}{x-k}$
$\displaystyle \frac{x^{3}+3x^{2}+11x+9}{x+1}=x^{2}+2x+9+\frac{0}{x+1}=x^{2}+2x+9$
