Divide using synthetic division $$ ( x^{4}-14x^{2}+5x ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&1&0&-14&5&0\\& & -4& 16& -8& \color{black}{12} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{2}&\color{blue}{-3}&\color{orangered}{12} \end{array} $$The solution is:
$$ \frac{ x^{4}-14x^{2}+5x }{ x+4 } = \color{blue}{x^{3}-4x^{2}+2x-3} ~+~ \frac{ \color{red}{ 12 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&0&-14&5&0\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 1 }&0&-14&5&0\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&0&-14&5&0\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}-4&1&\color{orangered}{ 0 }&-14&5&0\\& & \color{orangered}{-4} & & & \\ \hline &1&\color{orangered}{-4}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&0&-14&5&0\\& & -4& \color{blue}{16} & & \\ \hline &1&\color{blue}{-4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 16 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}-4&1&0&\color{orangered}{ -14 }&5&0\\& & -4& \color{orangered}{16} & & \\ \hline &1&-4&\color{orangered}{2}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 2 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&0&-14&5&0\\& & -4& 16& \color{blue}{-8} & \\ \hline &1&-4&\color{blue}{2}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}-4&1&0&-14&\color{orangered}{ 5 }&0\\& & -4& 16& \color{orangered}{-8} & \\ \hline &1&-4&2&\color{orangered}{-3}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&0&-14&5&0\\& & -4& 16& -8& \color{blue}{12} \\ \hline &1&-4&2&\color{blue}{-3}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 12 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrr}-4&1&0&-14&5&\color{orangered}{ 0 }\\& & -4& 16& -8& \color{orangered}{12} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{2}&\color{blue}{-3}&\color{orangered}{12} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-4x^{2}+2x-3 } $ with a remainder of $ \color{red}{ 12 } $.