Divide using synthetic division $$ ( 2x^{4}-7x^{3}-17x^{2}+59x-24 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&2&-7&-17&59&-24\\& & -6& 39& -66& \color{black}{21} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{blue}{22}&\color{blue}{-7}&\color{orangered}{-3} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-7x^{3}-17x^{2}+59x-24 }{ x+3 } = \color{blue}{2x^{3}-13x^{2}+22x-7} \color{red}{~-~} \frac{ \color{red}{ 3 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&-7&-17&59&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 2 }&-7&-17&59&-24\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&-7&-17&59&-24\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrrrr}-3&2&\color{orangered}{ -7 }&-17&59&-24\\& & \color{orangered}{-6} & & & \\ \hline &2&\color{orangered}{-13}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 39 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&-7&-17&59&-24\\& & -6& \color{blue}{39} & & \\ \hline &2&\color{blue}{-13}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 39 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrrrr}-3&2&-7&\color{orangered}{ -17 }&59&-24\\& & -6& \color{orangered}{39} & & \\ \hline &2&-13&\color{orangered}{22}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 22 } = \color{blue}{ -66 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&-7&-17&59&-24\\& & -6& 39& \color{blue}{-66} & \\ \hline &2&-13&\color{blue}{22}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 59 } + \color{orangered}{ \left( -66 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrr}-3&2&-7&-17&\color{orangered}{ 59 }&-24\\& & -6& 39& \color{orangered}{-66} & \\ \hline &2&-13&22&\color{orangered}{-7}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&2&-7&-17&59&-24\\& & -6& 39& -66& \color{blue}{21} \\ \hline &2&-13&22&\color{blue}{-7}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 21 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}-3&2&-7&-17&59&\color{orangered}{ -24 }\\& & -6& 39& -66& \color{orangered}{21} \\ \hline &\color{blue}{2}&\color{blue}{-13}&\color{blue}{22}&\color{blue}{-7}&\color{orangered}{-3} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-13x^{2}+22x-7 } $ with a remainder of $ \color{red}{ -3 } $.