Divide using synthetic division $$ ( 2x^{4}-7x^{3}-17x^{2}+59x-24 ) \div ( x-24 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}24&2&-7&-17&59&-24\\& & 48& 984& 23208& \color{black}{558408} \\ \hline &\color{blue}{2}&\color{blue}{41}&\color{blue}{967}&\color{blue}{23267}&\color{orangered}{558384} \end{array} $$The solution is:
$$ \dfrac{ 2x^{4}-7x^{3}-17x^{2}+59x-24 }{ x-24 } = \color{blue}{2x^{3}+41x^{2}+967x+23267} ~+~ \dfrac{ \color{red}{ 558384 } }{ x-24 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -24 = 0 $ ( $ x = \color{blue}{ 24 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{24}&2&-7&-17&59&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}24&\color{orangered}{ 2 }&-7&-17&59&-24\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 24 } \cdot \color{blue}{ 2 } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{24}&2&-7&-17&59&-24\\& & \color{blue}{48} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 48 } = \color{orangered}{ 41 } $
$$ \begin{array}{c|rrrrr}24&2&\color{orangered}{ -7 }&-17&59&-24\\& & \color{orangered}{48} & & & \\ \hline &2&\color{orangered}{41}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 24 } \cdot \color{blue}{ 41 } = \color{blue}{ 984 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{24}&2&-7&-17&59&-24\\& & 48& \color{blue}{984} & & \\ \hline &2&\color{blue}{41}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 984 } = \color{orangered}{ 967 } $
$$ \begin{array}{c|rrrrr}24&2&-7&\color{orangered}{ -17 }&59&-24\\& & 48& \color{orangered}{984} & & \\ \hline &2&41&\color{orangered}{967}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 24 } \cdot \color{blue}{ 967 } = \color{blue}{ 23208 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{24}&2&-7&-17&59&-24\\& & 48& 984& \color{blue}{23208} & \\ \hline &2&41&\color{blue}{967}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 59 } + \color{orangered}{ 23208 } = \color{orangered}{ 23267 } $
$$ \begin{array}{c|rrrrr}24&2&-7&-17&\color{orangered}{ 59 }&-24\\& & 48& 984& \color{orangered}{23208} & \\ \hline &2&41&967&\color{orangered}{23267}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 24 } \cdot \color{blue}{ 23267 } = \color{blue}{ 558408 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{24}&2&-7&-17&59&-24\\& & 48& 984& 23208& \color{blue}{558408} \\ \hline &2&41&967&\color{blue}{23267}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 558408 } = \color{orangered}{ 558384 } $
$$ \begin{array}{c|rrrrr}24&2&-7&-17&59&\color{orangered}{ -24 }\\& & 48& 984& 23208& \color{orangered}{558408} \\ \hline &\color{blue}{2}&\color{blue}{41}&\color{blue}{967}&\color{blue}{23267}&\color{orangered}{558384} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}+41x^{2}+967x+23267 } $ with a remainder of $ \color{red}{ 558384 } $.