Divide using synthetic division $$ ( 2x^{4}-7x^{3}-17x^{2}+59x-24 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-1&2&-7&-17&59&-24\\& & -2& 9& 8& \color{black}{-67} \\ \hline &\color{blue}{2}&\color{blue}{-9}&\color{blue}{-8}&\color{blue}{67}&\color{orangered}{-91} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-7x^{3}-17x^{2}+59x-24 }{ x+1 } = \color{blue}{2x^{3}-9x^{2}-8x+67} \color{red}{~-~} \frac{ \color{red}{ 91 } }{ x+1 } $$Explanation
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.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&2&-7&-17&59&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-1&\color{orangered}{ 2 }&-7&-17&59&-24\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 2 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&2&-7&-17&59&-24\\& & \color{blue}{-2} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-1&2&\color{orangered}{ -7 }&-17&59&-24\\& & \color{orangered}{-2} & & & \\ \hline &2&\color{orangered}{-9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&2&-7&-17&59&-24\\& & -2& \color{blue}{9} & & \\ \hline &2&\color{blue}{-9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 9 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}-1&2&-7&\color{orangered}{ -17 }&59&-24\\& & -2& \color{orangered}{9} & & \\ \hline &2&-9&\color{orangered}{-8}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&2&-7&-17&59&-24\\& & -2& 9& \color{blue}{8} & \\ \hline &2&-9&\color{blue}{-8}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 59 } + \color{orangered}{ 8 } = \color{orangered}{ 67 } $
$$ \begin{array}{c|rrrrr}-1&2&-7&-17&\color{orangered}{ 59 }&-24\\& & -2& 9& \color{orangered}{8} & \\ \hline &2&-9&-8&\color{orangered}{67}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 67 } = \color{blue}{ -67 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&2&-7&-17&59&-24\\& & -2& 9& 8& \color{blue}{-67} \\ \hline &2&-9&-8&\color{blue}{67}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ \left( -67 \right) } = \color{orangered}{ -91 } $
$$ \begin{array}{c|rrrrr}-1&2&-7&-17&59&\color{orangered}{ -24 }\\& & -2& 9& 8& \color{orangered}{-67} \\ \hline &\color{blue}{2}&\color{blue}{-9}&\color{blue}{-8}&\color{blue}{67}&\color{orangered}{-91} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-9x^{2}-8x+67 } $ with a remainder of $ \color{red}{ -91 } $.