Divide using synthetic division $$ ( x^{4}-5x^{3}+7x^{2}-34x-1 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&1&-5&7&-34&-1\\& & -3& 24& -93& \color{black}{381} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{31}&\color{blue}{-127}&\color{orangered}{380} \end{array} $$The solution is:
$$ \frac{ x^{4}-5x^{3}+7x^{2}-34x-1 }{ x+3 } = \color{blue}{x^{3}-8x^{2}+31x-127} ~+~ \frac{ \color{red}{ 380 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-5&7&-34&-1\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&-5&7&-34&-1\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-5&7&-34&-1\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ -5 }&7&-34&-1\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{-8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-5&7&-34&-1\\& & -3& \color{blue}{24} & & \\ \hline &1&\color{blue}{-8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 24 } = \color{orangered}{ 31 } $
$$ \begin{array}{c|rrrrr}-3&1&-5&\color{orangered}{ 7 }&-34&-1\\& & -3& \color{orangered}{24} & & \\ \hline &1&-8&\color{orangered}{31}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 31 } = \color{blue}{ -93 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-5&7&-34&-1\\& & -3& 24& \color{blue}{-93} & \\ \hline &1&-8&\color{blue}{31}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ \left( -93 \right) } = \color{orangered}{ -127 } $
$$ \begin{array}{c|rrrrr}-3&1&-5&7&\color{orangered}{ -34 }&-1\\& & -3& 24& \color{orangered}{-93} & \\ \hline &1&-8&31&\color{orangered}{-127}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -127 \right) } = \color{blue}{ 381 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-5&7&-34&-1\\& & -3& 24& -93& \color{blue}{381} \\ \hline &1&-8&31&\color{blue}{-127}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 381 } = \color{orangered}{ 380 } $
$$ \begin{array}{c|rrrrr}-3&1&-5&7&-34&\color{orangered}{ -1 }\\& & -3& 24& -93& \color{orangered}{381} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{31}&\color{blue}{-127}&\color{orangered}{380} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-8x^{2}+31x-127 } $ with a remainder of $ \color{red}{ 380 } $.