Divide using synthetic division $$ ( x^{4}-6x^{3}+5x^{2}+30x-50 ) \div ( -5x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&1&-6&5&30&-50\\& & -2& 16& -42& \color{black}{24} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{21}&\color{blue}{-12}&\color{orangered}{-26} \end{array} $$The solution is:
$$ \frac{ x^{4}-6x^{3}+5x^{2}+30x-50 }{ x+2 } = \color{blue}{x^{3}-8x^{2}+21x-12} \color{red}{~-~} \frac{ \color{red}{ 26 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-6&5&30&-50\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 1 }&-6&5&30&-50\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-6&5&30&-50\\& & \color{blue}{-2} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}-2&1&\color{orangered}{ -6 }&5&30&-50\\& & \color{orangered}{-2} & & & \\ \hline &1&\color{orangered}{-8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-6&5&30&-50\\& & -2& \color{blue}{16} & & \\ \hline &1&\color{blue}{-8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 16 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrrr}-2&1&-6&\color{orangered}{ 5 }&30&-50\\& & -2& \color{orangered}{16} & & \\ \hline &1&-8&\color{orangered}{21}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 21 } = \color{blue}{ -42 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-6&5&30&-50\\& & -2& 16& \color{blue}{-42} & \\ \hline &1&-8&\color{blue}{21}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -42 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrr}-2&1&-6&5&\color{orangered}{ 30 }&-50\\& & -2& 16& \color{orangered}{-42} & \\ \hline &1&-8&21&\color{orangered}{-12}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-6&5&30&-50\\& & -2& 16& -42& \color{blue}{24} \\ \hline &1&-8&21&\color{blue}{-12}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -50 } + \color{orangered}{ 24 } = \color{orangered}{ -26 } $
$$ \begin{array}{c|rrrrr}-2&1&-6&5&30&\color{orangered}{ -50 }\\& & -2& 16& -42& \color{orangered}{24} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{21}&\color{blue}{-12}&\color{orangered}{-26} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-8x^{2}+21x-12 } $ with a remainder of $ \color{red}{ -26 } $.