Divide using synthetic division $$ ( 5x^{4}-3x^{3}+2x^{2}-2 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&5&-3&2&0&-2\\& & -20& 92& -376& \color{black}{1504} \\ \hline &\color{blue}{5}&\color{blue}{-23}&\color{blue}{94}&\color{blue}{-376}&\color{orangered}{1502} \end{array} $$The solution is:
$$ \frac{ 5x^{4}-3x^{3}+2x^{2}-2 }{ x+4 } = \color{blue}{5x^{3}-23x^{2}+94x-376} ~+~ \frac{ \color{red}{ 1502 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&5&-3&2&0&-2\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 5 }&-3&2&0&-2\\& & & & & \\ \hline &\color{orangered}{5}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 5 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&5&-3&2&0&-2\\& & \color{blue}{-20} & & & \\ \hline &\color{blue}{5}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -23 } $
$$ \begin{array}{c|rrrrr}-4&5&\color{orangered}{ -3 }&2&0&-2\\& & \color{orangered}{-20} & & & \\ \hline &5&\color{orangered}{-23}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -23 \right) } = \color{blue}{ 92 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&5&-3&2&0&-2\\& & -20& \color{blue}{92} & & \\ \hline &5&\color{blue}{-23}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 92 } = \color{orangered}{ 94 } $
$$ \begin{array}{c|rrrrr}-4&5&-3&\color{orangered}{ 2 }&0&-2\\& & -20& \color{orangered}{92} & & \\ \hline &5&-23&\color{orangered}{94}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 94 } = \color{blue}{ -376 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&5&-3&2&0&-2\\& & -20& 92& \color{blue}{-376} & \\ \hline &5&-23&\color{blue}{94}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -376 \right) } = \color{orangered}{ -376 } $
$$ \begin{array}{c|rrrrr}-4&5&-3&2&\color{orangered}{ 0 }&-2\\& & -20& 92& \color{orangered}{-376} & \\ \hline &5&-23&94&\color{orangered}{-376}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -376 \right) } = \color{blue}{ 1504 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&5&-3&2&0&-2\\& & -20& 92& -376& \color{blue}{1504} \\ \hline &5&-23&94&\color{blue}{-376}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 1504 } = \color{orangered}{ 1502 } $
$$ \begin{array}{c|rrrrr}-4&5&-3&2&0&\color{orangered}{ -2 }\\& & -20& 92& -376& \color{orangered}{1504} \\ \hline &\color{blue}{5}&\color{blue}{-23}&\color{blue}{94}&\color{blue}{-376}&\color{orangered}{1502} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{3}-23x^{2}+94x-376 } $ with a remainder of $ \color{red}{ 1502 } $.