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