Answer
$5x^{3}+2x-3$
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.
$k=-1$
$ \begin{array}{lllllll}
& & -- & -- & -- & -- & --\\
-1 & ) & 5 & 5 & 2 & -1 & -3\\
& & & & & & \\
& & -- & -- & -- & -- & --\\
& & & & & & \\
& & & & & &
\end{array}$
We are ready.
Bring down the leading coefficient, $5$
Multiply $(-1)(5)=-5$ and place it in the next free slot of the middle row.
$ \begin{array}{lllllll}
& & -- & -- & -- & -- & --\\
-1 & ) & 5 & 5 & 2 & -1 & -3\\
& & & -5 & & & \\
& & -- & -- & -- & -- & --\\
& & 5 & & & & \\
& & & & & &
\end{array}$
Add $5+(-5)=0$ and place $0$ in the bottom row.
Multiply $(-1)(0)=0$ and place it in the next free slot of the middle row.
$ \begin{array}{lllllll}
& & -- & -- & -- & -- & --\\
-1 & ) & 5 & 5 & 2 & -1 & -3\\
& & & -5 & 0 & & \\
& & -- & -- & -- & -- & --\\
& & 5 & 0 & & & \\
& & & & & &
\end{array}$
Add $2+(0) =2$ 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 & ) & 5 & 5 & 2 & -1 & -3\\
& & & -5 & 0 & -2 & \\
& & -- & -- & -- & -- & --\\
& & 5 & 0 & 2 & & \\
& & & & & &
\end{array}$
Add $-1+(-2)=-3$ and place it in the bottom row.
Multiply $(-1)(-3)=3$ and place it in the next free slot of the middle row.
$ \begin{array}{lllllll}
& & -- & -- & -- & -- & --\\
-1 & ) & 5 & 5 & 2 & -1 & -3\\
& & & -5 & 0 & -2 & 3\\
& & -- & -- & -- & -- & --\\
& & 5 & 0 & 2 & -3 & \\
& & & & & &
\end{array}$
Add $-3+(3)=0$ and place it in the bottom row.
$ \begin{array}{lllllll}
& & -- & -- & -- & -- & --\\
-1 & ) & 5 & 5 & 2 & -1 & -3\\
& & & -5 & 0 & -2 & 3\\
& & -- & -- & -- & -- & --\\
& & 5 & 0 & 2 & -3 & \fbox{$0$}\\
& & & & & &
\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 $f(x):\quad q(x)=5x^{3}+2x-3$
So, $\displaystyle \quad \frac{f(x)}{x-k}=q(x)+\frac{r}{x-k}$
$\displaystyle \frac{5x^{4}+5x^{3}+2x^{2}-x-3}{x+1}=\quad 5x^{3}+2x-3+\frac{0}{x+1}\quad =5x^{3}+2x-3$