Divide using synthetic division $$ ( 3x^{3}-16x^{2}+17x-4 ) \div ( x+1 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&3&-16&17&-4\\& & -3& 19& \color{black}{-36} \\ \hline &\color{blue}{3}&\color{blue}{-19}&\color{blue}{36}&\color{orangered}{-40} \end{array} $$The solution is:
$$ \dfrac{ 3x^{3}-16x^{2}+17x-4 }{ x+1 } = \color{blue}{3x^{2}-19x+36} \color{red}{~-~} \dfrac{ \color{red}{ 40 } }{ x+1 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-16&17&-4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 3 }&-16&17&-4\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 3 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-16&17&-4\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrrr}-1&3&\color{orangered}{ -16 }&17&-4\\& & \color{orangered}{-3} & & \\ \hline &3&\color{orangered}{-19}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ 19 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-16&17&-4\\& & -3& \color{blue}{19} & \\ \hline &3&\color{blue}{-19}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 19 } = \color{orangered}{ 36 } $
$$ \begin{array}{c|rrrr}-1&3&-16&\color{orangered}{ 17 }&-4\\& & -3& \color{orangered}{19} & \\ \hline &3&-19&\color{orangered}{36}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 36 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&3&-16&17&-4\\& & -3& 19& \color{blue}{-36} \\ \hline &3&-19&\color{blue}{36}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -40 } $
$$ \begin{array}{c|rrrr}-1&3&-16&17&\color{orangered}{ -4 }\\& & -3& 19& \color{orangered}{-36} \\ \hline &\color{blue}{3}&\color{blue}{-19}&\color{blue}{36}&\color{orangered}{-40} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-19x+36 } $ with a remainder of $ \color{red}{ -40 } $.