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