Divide using synthetic division $$ ( 2x^{4}-7x^{3}-17x^{2}+59x-24 ) \div ( x+4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&2&-7&-17&59&-24\\& & -8& 60& -172& \color{black}{452} \\ \hline &\color{blue}{2}&\color{blue}{-15}&\color{blue}{43}&\color{blue}{-113}&\color{orangered}{428} \end{array} $$The solution is:
$$ \dfrac{ 2x^{4}-7x^{3}-17x^{2}+59x-24 }{ x+4 } = \color{blue}{2x^{3}-15x^{2}+43x-113} ~+~ \dfrac{ \color{red}{ 428 } }{ x+4 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&2&-7&-17&59&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 2 }&-7&-17&59&-24\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 2 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&2&-7&-17&59&-24\\& & \color{blue}{-8} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrrr}-4&2&\color{orangered}{ -7 }&-17&59&-24\\& & \color{orangered}{-8} & & & \\ \hline &2&\color{orangered}{-15}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&2&-7&-17&59&-24\\& & -8& \color{blue}{60} & & \\ \hline &2&\color{blue}{-15}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 60 } = \color{orangered}{ 43 } $
$$ \begin{array}{c|rrrrr}-4&2&-7&\color{orangered}{ -17 }&59&-24\\& & -8& \color{orangered}{60} & & \\ \hline &2&-15&\color{orangered}{43}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 43 } = \color{blue}{ -172 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&2&-7&-17&59&-24\\& & -8& 60& \color{blue}{-172} & \\ \hline &2&-15&\color{blue}{43}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 59 } + \color{orangered}{ \left( -172 \right) } = \color{orangered}{ -113 } $
$$ \begin{array}{c|rrrrr}-4&2&-7&-17&\color{orangered}{ 59 }&-24\\& & -8& 60& \color{orangered}{-172} & \\ \hline &2&-15&43&\color{orangered}{-113}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -113 \right) } = \color{blue}{ 452 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&2&-7&-17&59&-24\\& & -8& 60& -172& \color{blue}{452} \\ \hline &2&-15&43&\color{blue}{-113}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 452 } = \color{orangered}{ 428 } $
$$ \begin{array}{c|rrrrr}-4&2&-7&-17&59&\color{orangered}{ -24 }\\& & -8& 60& -172& \color{orangered}{452} \\ \hline &\color{blue}{2}&\color{blue}{-15}&\color{blue}{43}&\color{blue}{-113}&\color{orangered}{428} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-15x^{2}+43x-113 } $ with a remainder of $ \color{red}{ 428 } $.