Divide using synthetic division $$ ( 5x^{6}-4x^{5}-19x^{4}-9x^{3}+9x^{2}+2x-5 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-1&5&-4&-19&-9&9&2&-5\\& & -5& 9& 10& -1& -8& \color{black}{6} \\ \hline &\color{blue}{5}&\color{blue}{-9}&\color{blue}{-10}&\color{blue}{1}&\color{blue}{8}&\color{blue}{-6}&\color{orangered}{1} \end{array} $$The solution is:
$$ \frac{ 5x^{6}-4x^{5}-19x^{4}-9x^{3}+9x^{2}+2x-5 }{ x+1 } = \color{blue}{5x^{5}-9x^{4}-10x^{3}+x^{2}+8x-6} ~+~ \frac{ \color{red}{ 1 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&5&-4&-19&-9&9&2&-5\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-1&\color{orangered}{ 5 }&-4&-19&-9&9&2&-5\\& & & & & & & \\ \hline &\color{orangered}{5}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 5 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&5&-4&-19&-9&9&2&-5\\& & \color{blue}{-5} & & & & & \\ \hline &\color{blue}{5}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrrrr}-1&5&\color{orangered}{ -4 }&-19&-9&9&2&-5\\& & \color{orangered}{-5} & & & & & \\ \hline &5&\color{orangered}{-9}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&5&-4&-19&-9&9&2&-5\\& & -5& \color{blue}{9} & & & & \\ \hline &5&\color{blue}{-9}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 9 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrrrr}-1&5&-4&\color{orangered}{ -19 }&-9&9&2&-5\\& & -5& \color{orangered}{9} & & & & \\ \hline &5&-9&\color{orangered}{-10}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&5&-4&-19&-9&9&2&-5\\& & -5& 9& \color{blue}{10} & & & \\ \hline &5&-9&\color{blue}{-10}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 10 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrrr}-1&5&-4&-19&\color{orangered}{ -9 }&9&2&-5\\& & -5& 9& \color{orangered}{10} & & & \\ \hline &5&-9&-10&\color{orangered}{1}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 1 } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&5&-4&-19&-9&9&2&-5\\& & -5& 9& 10& \color{blue}{-1} & & \\ \hline &5&-9&-10&\color{blue}{1}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrrr}-1&5&-4&-19&-9&\color{orangered}{ 9 }&2&-5\\& & -5& 9& 10& \color{orangered}{-1} & & \\ \hline &5&-9&-10&1&\color{orangered}{8}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 8 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&5&-4&-19&-9&9&2&-5\\& & -5& 9& 10& -1& \color{blue}{-8} & \\ \hline &5&-9&-10&1&\color{blue}{8}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrrr}-1&5&-4&-19&-9&9&\color{orangered}{ 2 }&-5\\& & -5& 9& 10& -1& \color{orangered}{-8} & \\ \hline &5&-9&-10&1&8&\color{orangered}{-6}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&5&-4&-19&-9&9&2&-5\\& & -5& 9& 10& -1& -8& \color{blue}{6} \\ \hline &5&-9&-10&1&8&\color{blue}{-6}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 6 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrrr}-1&5&-4&-19&-9&9&2&\color{orangered}{ -5 }\\& & -5& 9& 10& -1& -8& \color{orangered}{6} \\ \hline &\color{blue}{5}&\color{blue}{-9}&\color{blue}{-10}&\color{blue}{1}&\color{blue}{8}&\color{blue}{-6}&\color{orangered}{1} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{5}-9x^{4}-10x^{3}+x^{2}+8x-6 } $ with a remainder of $ \color{red}{ 1 } $.