The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&17&0&-11&-6\\& & -34& 68& \color{black}{-114} \\ \hline &\color{blue}{17}&\color{blue}{-34}&\color{blue}{57}&\color{orangered}{-120} \end{array} $$The solution is:
$$ \frac{ 17x^{3}-11x-6 }{ x+2 } = \color{blue}{17x^{2}-34x+57} \color{red}{~-~} \frac{ \color{red}{ 120 } }{ x+2 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&17&0&-11&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 17 }&0&-11&-6\\& & & & \\ \hline &\color{orangered}{17}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 17 } = \color{blue}{ -34 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&17&0&-11&-6\\& & \color{blue}{-34} & & \\ \hline &\color{blue}{17}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -34 \right) } = \color{orangered}{ -34 } $
$$ \begin{array}{c|rrrr}-2&17&\color{orangered}{ 0 }&-11&-6\\& & \color{orangered}{-34} & & \\ \hline &17&\color{orangered}{-34}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -34 \right) } = \color{blue}{ 68 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&17&0&-11&-6\\& & -34& \color{blue}{68} & \\ \hline &17&\color{blue}{-34}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 68 } = \color{orangered}{ 57 } $
$$ \begin{array}{c|rrrr}-2&17&0&\color{orangered}{ -11 }&-6\\& & -34& \color{orangered}{68} & \\ \hline &17&-34&\color{orangered}{57}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 57 } = \color{blue}{ -114 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&17&0&-11&-6\\& & -34& 68& \color{blue}{-114} \\ \hline &17&-34&\color{blue}{57}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -114 \right) } = \color{orangered}{ -120 } $
$$ \begin{array}{c|rrrr}-2&17&0&-11&\color{orangered}{ -6 }\\& & -34& 68& \color{orangered}{-114} \\ \hline &\color{blue}{17}&\color{blue}{-34}&\color{blue}{57}&\color{orangered}{-120} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 17x^{2}-34x+57 } $ with a remainder of $ \color{red}{ -120 } $.