Divide using synthetic division $$ ( x^{4}-3x^{3}+5x-2 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&1&-3&0&5&-2\\& & -2& 10& -20& \color{black}{30} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{10}&\color{blue}{-15}&\color{orangered}{28} \end{array} $$The solution is:
$$ \frac{ x^{4}-3x^{3}+5x-2 }{ x+2 } = \color{blue}{x^{3}-5x^{2}+10x-15} ~+~ \frac{ \color{red}{ 28 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-3&0&5&-2\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 1 }&-3&0&5&-2\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-3&0&5&-2\\& & \color{blue}{-2} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrr}-2&1&\color{orangered}{ -3 }&0&5&-2\\& & \color{orangered}{-2} & & & \\ \hline &1&\color{orangered}{-5}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-3&0&5&-2\\& & -2& \color{blue}{10} & & \\ \hline &1&\color{blue}{-5}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 10 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrr}-2&1&-3&\color{orangered}{ 0 }&5&-2\\& & -2& \color{orangered}{10} & & \\ \hline &1&-5&\color{orangered}{10}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 10 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-3&0&5&-2\\& & -2& 10& \color{blue}{-20} & \\ \hline &1&-5&\color{blue}{10}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrrr}-2&1&-3&0&\color{orangered}{ 5 }&-2\\& & -2& 10& \color{orangered}{-20} & \\ \hline &1&-5&10&\color{orangered}{-15}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-3&0&5&-2\\& & -2& 10& -20& \color{blue}{30} \\ \hline &1&-5&10&\color{blue}{-15}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 30 } = \color{orangered}{ 28 } $
$$ \begin{array}{c|rrrrr}-2&1&-3&0&5&\color{orangered}{ -2 }\\& & -2& 10& -20& \color{orangered}{30} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{10}&\color{blue}{-15}&\color{orangered}{28} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-5x^{2}+10x-15 } $ with a remainder of $ \color{red}{ 28 } $.