Divide using synthetic division $$ ( 4x^{4}-20x^{3}+33x^{2}-20x+4 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-1&4&-20&33&-20&4\\& & -4& 24& -57& \color{black}{77} \\ \hline &\color{blue}{4}&\color{blue}{-24}&\color{blue}{57}&\color{blue}{-77}&\color{orangered}{81} \end{array} $$The solution is:
$$ \frac{ 4x^{4}-20x^{3}+33x^{2}-20x+4 }{ x+1 } = \color{blue}{4x^{3}-24x^{2}+57x-77} ~+~ \frac{ \color{red}{ 81 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&-20&33&-20&4\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-1&\color{orangered}{ 4 }&-20&33&-20&4\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 4 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&-20&33&-20&4\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrrr}-1&4&\color{orangered}{ -20 }&33&-20&4\\& & \color{orangered}{-4} & & & \\ \hline &4&\color{orangered}{-24}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&-20&33&-20&4\\& & -4& \color{blue}{24} & & \\ \hline &4&\color{blue}{-24}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 33 } + \color{orangered}{ 24 } = \color{orangered}{ 57 } $
$$ \begin{array}{c|rrrrr}-1&4&-20&\color{orangered}{ 33 }&-20&4\\& & -4& \color{orangered}{24} & & \\ \hline &4&-24&\color{orangered}{57}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 57 } = \color{blue}{ -57 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&-20&33&-20&4\\& & -4& 24& \color{blue}{-57} & \\ \hline &4&-24&\color{blue}{57}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ \left( -57 \right) } = \color{orangered}{ -77 } $
$$ \begin{array}{c|rrrrr}-1&4&-20&33&\color{orangered}{ -20 }&4\\& & -4& 24& \color{orangered}{-57} & \\ \hline &4&-24&57&\color{orangered}{-77}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -77 \right) } = \color{blue}{ 77 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&4&-20&33&-20&4\\& & -4& 24& -57& \color{blue}{77} \\ \hline &4&-24&57&\color{blue}{-77}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 77 } = \color{orangered}{ 81 } $
$$ \begin{array}{c|rrrrr}-1&4&-20&33&-20&\color{orangered}{ 4 }\\& & -4& 24& -57& \color{orangered}{77} \\ \hline &\color{blue}{4}&\color{blue}{-24}&\color{blue}{57}&\color{blue}{-77}&\color{orangered}{81} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}-24x^{2}+57x-77 } $ with a remainder of $ \color{red}{ 81 } $.