Divide using synthetic division $$ ( 3x^{4}+2x^{3}-23x^{2}+4x+20 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-1&3&2&-23&4&20\\& & -3& 1& 22& \color{black}{-26} \\ \hline &\color{blue}{3}&\color{blue}{-1}&\color{blue}{-22}&\color{blue}{26}&\color{orangered}{-6} \end{array} $$The solution is:
$$ \frac{ 3x^{4}+2x^{3}-23x^{2}+4x+20 }{ x+1 } = \color{blue}{3x^{3}-x^{2}-22x+26} \color{red}{~-~} \frac{ \color{red}{ 6 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&2&-23&4&20\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-1&\color{orangered}{ 3 }&2&-23&4&20\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 3 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&2&-23&4&20\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}-1&3&\color{orangered}{ 2 }&-23&4&20\\& & \color{orangered}{-3} & & & \\ \hline &3&\color{orangered}{-1}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&2&-23&4&20\\& & -3& \color{blue}{1} & & \\ \hline &3&\color{blue}{-1}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -23 } + \color{orangered}{ 1 } = \color{orangered}{ -22 } $
$$ \begin{array}{c|rrrrr}-1&3&2&\color{orangered}{ -23 }&4&20\\& & -3& \color{orangered}{1} & & \\ \hline &3&-1&\color{orangered}{-22}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -22 \right) } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&2&-23&4&20\\& & -3& 1& \color{blue}{22} & \\ \hline &3&-1&\color{blue}{-22}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 22 } = \color{orangered}{ 26 } $
$$ \begin{array}{c|rrrrr}-1&3&2&-23&\color{orangered}{ 4 }&20\\& & -3& 1& \color{orangered}{22} & \\ \hline &3&-1&-22&\color{orangered}{26}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 26 } = \color{blue}{ -26 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&2&-23&4&20\\& & -3& 1& 22& \color{blue}{-26} \\ \hline &3&-1&-22&\color{blue}{26}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -26 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}-1&3&2&-23&4&\color{orangered}{ 20 }\\& & -3& 1& 22& \color{orangered}{-26} \\ \hline &\color{blue}{3}&\color{blue}{-1}&\color{blue}{-22}&\color{blue}{26}&\color{orangered}{-6} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-x^{2}-22x+26 } $ with a remainder of $ \color{red}{ -6 } $.