Divide using synthetic division $$ ( 2x^{4}-7x^{3}-17x^{2}+59x-24 ) \div ( x+6 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-6&2&-7&-17&59&-24\\& & -12& 114& -582& \color{black}{3138} \\ \hline &\color{blue}{2}&\color{blue}{-19}&\color{blue}{97}&\color{blue}{-523}&\color{orangered}{3114} \end{array} $$The solution is:
$$ \dfrac{ 2x^{4}-7x^{3}-17x^{2}+59x-24 }{ x+6 } = \color{blue}{2x^{3}-19x^{2}+97x-523} ~+~ \dfrac{ \color{red}{ 3114 } }{ x+6 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&2&-7&-17&59&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-6&\color{orangered}{ 2 }&-7&-17&59&-24\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 2 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&2&-7&-17&59&-24\\& & \color{blue}{-12} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrrrr}-6&2&\color{orangered}{ -7 }&-17&59&-24\\& & \color{orangered}{-12} & & & \\ \hline &2&\color{orangered}{-19}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ 114 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&2&-7&-17&59&-24\\& & -12& \color{blue}{114} & & \\ \hline &2&\color{blue}{-19}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 114 } = \color{orangered}{ 97 } $
$$ \begin{array}{c|rrrrr}-6&2&-7&\color{orangered}{ -17 }&59&-24\\& & -12& \color{orangered}{114} & & \\ \hline &2&-19&\color{orangered}{97}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 97 } = \color{blue}{ -582 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&2&-7&-17&59&-24\\& & -12& 114& \color{blue}{-582} & \\ \hline &2&-19&\color{blue}{97}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 59 } + \color{orangered}{ \left( -582 \right) } = \color{orangered}{ -523 } $
$$ \begin{array}{c|rrrrr}-6&2&-7&-17&\color{orangered}{ 59 }&-24\\& & -12& 114& \color{orangered}{-582} & \\ \hline &2&-19&97&\color{orangered}{-523}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -523 \right) } = \color{blue}{ 3138 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&2&-7&-17&59&-24\\& & -12& 114& -582& \color{blue}{3138} \\ \hline &2&-19&97&\color{blue}{-523}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 3138 } = \color{orangered}{ 3114 } $
$$ \begin{array}{c|rrrrr}-6&2&-7&-17&59&\color{orangered}{ -24 }\\& & -12& 114& -582& \color{orangered}{3138} \\ \hline &\color{blue}{2}&\color{blue}{-19}&\color{blue}{97}&\color{blue}{-523}&\color{orangered}{3114} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-19x^{2}+97x-523 } $ with a remainder of $ \color{red}{ 3114 } $.