Divide using synthetic division $$ ( 2x^{4}-7x^{3}-17x^{2}+59x-24 ) \div ( x+12 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-12&2&-7&-17&59&-24\\& & -24& 372& -4260& \color{black}{50412} \\ \hline &\color{blue}{2}&\color{blue}{-31}&\color{blue}{355}&\color{blue}{-4201}&\color{orangered}{50388} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-7x^{3}-17x^{2}+59x-24 }{ x+12 } = \color{blue}{2x^{3}-31x^{2}+355x-4201} ~+~ \frac{ \color{red}{ 50388 } }{ x+12 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 12 = 0 $ ( $ x = \color{blue}{ -12 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-12}&2&-7&-17&59&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-12&\color{orangered}{ 2 }&-7&-17&59&-24\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ 2 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-12}&2&-7&-17&59&-24\\& & \color{blue}{-24} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -31 } $
$$ \begin{array}{c|rrrrr}-12&2&\color{orangered}{ -7 }&-17&59&-24\\& & \color{orangered}{-24} & & & \\ \hline &2&\color{orangered}{-31}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ \left( -31 \right) } = \color{blue}{ 372 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-12}&2&-7&-17&59&-24\\& & -24& \color{blue}{372} & & \\ \hline &2&\color{blue}{-31}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 372 } = \color{orangered}{ 355 } $
$$ \begin{array}{c|rrrrr}-12&2&-7&\color{orangered}{ -17 }&59&-24\\& & -24& \color{orangered}{372} & & \\ \hline &2&-31&\color{orangered}{355}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ 355 } = \color{blue}{ -4260 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-12}&2&-7&-17&59&-24\\& & -24& 372& \color{blue}{-4260} & \\ \hline &2&-31&\color{blue}{355}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 59 } + \color{orangered}{ \left( -4260 \right) } = \color{orangered}{ -4201 } $
$$ \begin{array}{c|rrrrr}-12&2&-7&-17&\color{orangered}{ 59 }&-24\\& & -24& 372& \color{orangered}{-4260} & \\ \hline &2&-31&355&\color{orangered}{-4201}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -12 } \cdot \color{blue}{ \left( -4201 \right) } = \color{blue}{ 50412 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-12}&2&-7&-17&59&-24\\& & -24& 372& -4260& \color{blue}{50412} \\ \hline &2&-31&355&\color{blue}{-4201}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 50412 } = \color{orangered}{ 50388 } $
$$ \begin{array}{c|rrrrr}-12&2&-7&-17&59&\color{orangered}{ -24 }\\& & -24& 372& -4260& \color{orangered}{50412} \\ \hline &\color{blue}{2}&\color{blue}{-31}&\color{blue}{355}&\color{blue}{-4201}&\color{orangered}{50388} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-31x^{2}+355x-4201 } $ with a remainder of $ \color{red}{ 50388 } $.