Divide using synthetic division $$ ( 3x^{4}-x^{3}-4x^{2}-10x-8 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-1&3&-1&-4&-10&-8\\& & -3& 4& 0& \color{black}{10} \\ \hline &\color{blue}{3}&\color{blue}{-4}&\color{blue}{0}&\color{blue}{-10}&\color{orangered}{2} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-x^{3}-4x^{2}-10x-8 }{ x+1 } = \color{blue}{3x^{3}-4x^{2}-10} ~+~ \frac{ \color{red}{ 2 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&-1&-4&-10&-8\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-1&\color{orangered}{ 3 }&-1&-4&-10&-8\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&-1&-4&-10&-8\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}-1&3&\color{orangered}{ -1 }&-4&-10&-8\\& & \color{orangered}{-3} & & & \\ \hline &3&\color{orangered}{-4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&-1&-4&-10&-8\\& & -3& \color{blue}{4} & & \\ \hline &3&\color{blue}{-4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 4 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-1&3&-1&\color{orangered}{ -4 }&-10&-8\\& & -3& \color{orangered}{4} & & \\ \hline &3&-4&\color{orangered}{0}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&-1&-4&-10&-8\\& & -3& 4& \color{blue}{0} & \\ \hline &3&-4&\color{blue}{0}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 0 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrr}-1&3&-1&-4&\color{orangered}{ -10 }&-8\\& & -3& 4& \color{orangered}{0} & \\ \hline &3&-4&0&\color{orangered}{-10}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&-1&-4&-10&-8\\& & -3& 4& 0& \color{blue}{10} \\ \hline &3&-4&0&\color{blue}{-10}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 10 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}-1&3&-1&-4&-10&\color{orangered}{ -8 }\\& & -3& 4& 0& \color{orangered}{10} \\ \hline &\color{blue}{3}&\color{blue}{-4}&\color{blue}{0}&\color{blue}{-10}&\color{orangered}{2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-4x^{2}-10 } $ with a remainder of $ \color{red}{ 2 } $.