Divide using synthetic division $$ ( x^{3}+17x^{2}+70x-54 ) \div ( x+6 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&1&17&70&-54\\& & -6& -66& \color{black}{-24} \\ \hline &\color{blue}{1}&\color{blue}{11}&\color{blue}{4}&\color{orangered}{-78} \end{array} $$The solution is:
$$ \dfrac{ x^{3}+17x^{2}+70x-54 }{ x+6 } = \color{blue}{x^{2}+11x+4} \color{red}{~-~} \dfrac{ \color{red}{ 78 } }{ x+6 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&17&70&-54\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 1 }&17&70&-54\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 1 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&17&70&-54\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrr}-6&1&\color{orangered}{ 17 }&70&-54\\& & \color{orangered}{-6} & & \\ \hline &1&\color{orangered}{11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 11 } = \color{blue}{ -66 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&17&70&-54\\& & -6& \color{blue}{-66} & \\ \hline &1&\color{blue}{11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 70 } + \color{orangered}{ \left( -66 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-6&1&17&\color{orangered}{ 70 }&-54\\& & -6& \color{orangered}{-66} & \\ \hline &1&11&\color{orangered}{4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 4 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&17&70&-54\\& & -6& -66& \color{blue}{-24} \\ \hline &1&11&\color{blue}{4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -78 } $
$$ \begin{array}{c|rrrr}-6&1&17&70&\color{orangered}{ -54 }\\& & -6& -66& \color{orangered}{-24} \\ \hline &\color{blue}{1}&\color{blue}{11}&\color{blue}{4}&\color{orangered}{-78} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+11x+4 } $ with a remainder of $ \color{red}{ -78 } $.