Divide using synthetic division $$ ( 24x^{4}-24x^{3}-32x+32 ) \div ( 4x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&24&-24&0&-32&32\\& & 96& 288& 1152& \color{black}{4480} \\ \hline &\color{blue}{24}&\color{blue}{72}&\color{blue}{288}&\color{blue}{1120}&\color{orangered}{4512} \end{array} $$The solution is:
$$ \frac{ 24x^{4}-24x^{3}-32x+32 }{ x-4 } = \color{blue}{24x^{3}+72x^{2}+288x+1120} ~+~ \frac{ \color{red}{ 4512 } }{ x-4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&24&-24&0&-32&32\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 24 }&-24&0&-32&32\\& & & & & \\ \hline &\color{orangered}{24}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 24 } = \color{blue}{ 96 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&24&-24&0&-32&32\\& & \color{blue}{96} & & & \\ \hline &\color{blue}{24}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 96 } = \color{orangered}{ 72 } $
$$ \begin{array}{c|rrrrr}4&24&\color{orangered}{ -24 }&0&-32&32\\& & \color{orangered}{96} & & & \\ \hline &24&\color{orangered}{72}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 72 } = \color{blue}{ 288 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&24&-24&0&-32&32\\& & 96& \color{blue}{288} & & \\ \hline &24&\color{blue}{72}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 288 } = \color{orangered}{ 288 } $
$$ \begin{array}{c|rrrrr}4&24&-24&\color{orangered}{ 0 }&-32&32\\& & 96& \color{orangered}{288} & & \\ \hline &24&72&\color{orangered}{288}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 288 } = \color{blue}{ 1152 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&24&-24&0&-32&32\\& & 96& 288& \color{blue}{1152} & \\ \hline &24&72&\color{blue}{288}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 1152 } = \color{orangered}{ 1120 } $
$$ \begin{array}{c|rrrrr}4&24&-24&0&\color{orangered}{ -32 }&32\\& & 96& 288& \color{orangered}{1152} & \\ \hline &24&72&288&\color{orangered}{1120}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 1120 } = \color{blue}{ 4480 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&24&-24&0&-32&32\\& & 96& 288& 1152& \color{blue}{4480} \\ \hline &24&72&288&\color{blue}{1120}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 32 } + \color{orangered}{ 4480 } = \color{orangered}{ 4512 } $
$$ \begin{array}{c|rrrrr}4&24&-24&0&-32&\color{orangered}{ 32 }\\& & 96& 288& 1152& \color{orangered}{4480} \\ \hline &\color{blue}{24}&\color{blue}{72}&\color{blue}{288}&\color{blue}{1120}&\color{orangered}{4512} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 24x^{3}+72x^{2}+288x+1120 } $ with a remainder of $ \color{red}{ 4512 } $.