Divide using synthetic division $$ ( x^{5}+19x^{4}-2x^{3}+7x^{2}-11x-28 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-4&1&19&-2&7&-11&-28\\& & -4& -60& 248& -1020& \color{black}{4124} \\ \hline &\color{blue}{1}&\color{blue}{15}&\color{blue}{-62}&\color{blue}{255}&\color{blue}{-1031}&\color{orangered}{4096} \end{array} $$The solution is:
$$ \frac{ x^{5}+19x^{4}-2x^{3}+7x^{2}-11x-28 }{ x+4 } = \color{blue}{x^{4}+15x^{3}-62x^{2}+255x-1031} ~+~ \frac{ \color{red}{ 4096 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&19&-2&7&-11&-28\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-4&\color{orangered}{ 1 }&19&-2&7&-11&-28\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&19&-2&7&-11&-28\\& & \color{blue}{-4} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrrrr}-4&1&\color{orangered}{ 19 }&-2&7&-11&-28\\& & \color{orangered}{-4} & & & & \\ \hline &1&\color{orangered}{15}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 15 } = \color{blue}{ -60 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&19&-2&7&-11&-28\\& & -4& \color{blue}{-60} & & & \\ \hline &1&\color{blue}{15}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -60 \right) } = \color{orangered}{ -62 } $
$$ \begin{array}{c|rrrrrr}-4&1&19&\color{orangered}{ -2 }&7&-11&-28\\& & -4& \color{orangered}{-60} & & & \\ \hline &1&15&\color{orangered}{-62}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -62 \right) } = \color{blue}{ 248 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&19&-2&7&-11&-28\\& & -4& -60& \color{blue}{248} & & \\ \hline &1&15&\color{blue}{-62}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 248 } = \color{orangered}{ 255 } $
$$ \begin{array}{c|rrrrrr}-4&1&19&-2&\color{orangered}{ 7 }&-11&-28\\& & -4& -60& \color{orangered}{248} & & \\ \hline &1&15&-62&\color{orangered}{255}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 255 } = \color{blue}{ -1020 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&19&-2&7&-11&-28\\& & -4& -60& 248& \color{blue}{-1020} & \\ \hline &1&15&-62&\color{blue}{255}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ \left( -1020 \right) } = \color{orangered}{ -1031 } $
$$ \begin{array}{c|rrrrrr}-4&1&19&-2&7&\color{orangered}{ -11 }&-28\\& & -4& -60& 248& \color{orangered}{-1020} & \\ \hline &1&15&-62&255&\color{orangered}{-1031}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -1031 \right) } = \color{blue}{ 4124 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&19&-2&7&-11&-28\\& & -4& -60& 248& -1020& \color{blue}{4124} \\ \hline &1&15&-62&255&\color{blue}{-1031}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 4124 } = \color{orangered}{ 4096 } $
$$ \begin{array}{c|rrrrrr}-4&1&19&-2&7&-11&\color{orangered}{ -28 }\\& & -4& -60& 248& -1020& \color{orangered}{4124} \\ \hline &\color{blue}{1}&\color{blue}{15}&\color{blue}{-62}&\color{blue}{255}&\color{blue}{-1031}&\color{orangered}{4096} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+15x^{3}-62x^{2}+255x-1031 } $ with a remainder of $ \color{red}{ 4096 } $.