Divide using synthetic division $$ ( 2x^{4}-7x^{3}-17x^{2}+59x-24 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&2&-7&-17&59&-24\\& & -4& 22& -10& \color{black}{-98} \\ \hline &\color{blue}{2}&\color{blue}{-11}&\color{blue}{5}&\color{blue}{49}&\color{orangered}{-122} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-7x^{3}-17x^{2}+59x-24 }{ x+2 } = \color{blue}{2x^{3}-11x^{2}+5x+49} \color{red}{~-~} \frac{ \color{red}{ 122 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-7&-17&59&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 2 }&-7&-17&59&-24\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 2 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-7&-17&59&-24\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrrr}-2&2&\color{orangered}{ -7 }&-17&59&-24\\& & \color{orangered}{-4} & & & \\ \hline &2&\color{orangered}{-11}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-7&-17&59&-24\\& & -4& \color{blue}{22} & & \\ \hline &2&\color{blue}{-11}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 22 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrr}-2&2&-7&\color{orangered}{ -17 }&59&-24\\& & -4& \color{orangered}{22} & & \\ \hline &2&-11&\color{orangered}{5}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-7&-17&59&-24\\& & -4& 22& \color{blue}{-10} & \\ \hline &2&-11&\color{blue}{5}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 59 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ 49 } $
$$ \begin{array}{c|rrrrr}-2&2&-7&-17&\color{orangered}{ 59 }&-24\\& & -4& 22& \color{orangered}{-10} & \\ \hline &2&-11&5&\color{orangered}{49}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 49 } = \color{blue}{ -98 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-7&-17&59&-24\\& & -4& 22& -10& \color{blue}{-98} \\ \hline &2&-11&5&\color{blue}{49}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ \left( -98 \right) } = \color{orangered}{ -122 } $
$$ \begin{array}{c|rrrrr}-2&2&-7&-17&59&\color{orangered}{ -24 }\\& & -4& 22& -10& \color{orangered}{-98} \\ \hline &\color{blue}{2}&\color{blue}{-11}&\color{blue}{5}&\color{blue}{49}&\color{orangered}{-122} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-11x^{2}+5x+49 } $ with a remainder of $ \color{red}{ -122 } $.