Divide using synthetic division $$ ( x^{5}-5x^{3}+2x ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-4&1&0&-5&0&2&0\\& & -4& 16& -44& 176& \color{black}{-712} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{11}&\color{blue}{-44}&\color{blue}{178}&\color{orangered}{-712} \end{array} $$The solution is:
$$ \frac{ x^{5}-5x^{3}+2x }{ x+4 } = \color{blue}{x^{4}-4x^{3}+11x^{2}-44x+178} \color{red}{~-~} \frac{ \color{red}{ 712 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&0&-5&0&2&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-4&\color{orangered}{ 1 }&0&-5&0&2&0\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&0&-5&0&2&0\\& & \color{blue}{-4} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}-4&1&\color{orangered}{ 0 }&-5&0&2&0\\& & \color{orangered}{-4} & & & & \\ \hline &1&\color{orangered}{-4}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&0&-5&0&2&0\\& & -4& \color{blue}{16} & & & \\ \hline &1&\color{blue}{-4}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 16 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrrrr}-4&1&0&\color{orangered}{ -5 }&0&2&0\\& & -4& \color{orangered}{16} & & & \\ \hline &1&-4&\color{orangered}{11}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 11 } = \color{blue}{ -44 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&0&-5&0&2&0\\& & -4& 16& \color{blue}{-44} & & \\ \hline &1&-4&\color{blue}{11}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -44 \right) } = \color{orangered}{ -44 } $
$$ \begin{array}{c|rrrrrr}-4&1&0&-5&\color{orangered}{ 0 }&2&0\\& & -4& 16& \color{orangered}{-44} & & \\ \hline &1&-4&11&\color{orangered}{-44}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -44 \right) } = \color{blue}{ 176 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&0&-5&0&2&0\\& & -4& 16& -44& \color{blue}{176} & \\ \hline &1&-4&11&\color{blue}{-44}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 176 } = \color{orangered}{ 178 } $
$$ \begin{array}{c|rrrrrr}-4&1&0&-5&0&\color{orangered}{ 2 }&0\\& & -4& 16& -44& \color{orangered}{176} & \\ \hline &1&-4&11&-44&\color{orangered}{178}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 178 } = \color{blue}{ -712 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&0&-5&0&2&0\\& & -4& 16& -44& 176& \color{blue}{-712} \\ \hline &1&-4&11&-44&\color{blue}{178}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -712 \right) } = \color{orangered}{ -712 } $
$$ \begin{array}{c|rrrrrr}-4&1&0&-5&0&2&\color{orangered}{ 0 }\\& & -4& 16& -44& 176& \color{orangered}{-712} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{11}&\color{blue}{-44}&\color{blue}{178}&\color{orangered}{-712} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-4x^{3}+11x^{2}-44x+178 } $ with a remainder of $ \color{red}{ -712 } $.