Divide using synthetic division $$ ( 5x^{5}+19x^{4}-2x^{3}+7x^{2}-11x-28 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-5&5&19&-2&7&-11&-28\\& & -25& 30& -140& 665& \color{black}{-3270} \\ \hline &\color{blue}{5}&\color{blue}{-6}&\color{blue}{28}&\color{blue}{-133}&\color{blue}{654}&\color{orangered}{-3298} \end{array} $$The solution is:
$$ \frac{ 5x^{5}+19x^{4}-2x^{3}+7x^{2}-11x-28 }{ x+5 } = \color{blue}{5x^{4}-6x^{3}+28x^{2}-133x+654} \color{red}{~-~} \frac{ \color{red}{ 3298 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&5&19&-2&7&-11&-28\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-5&\color{orangered}{ 5 }&19&-2&7&-11&-28\\& & & & & & \\ \hline &\color{orangered}{5}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 5 } = \color{blue}{ -25 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&5&19&-2&7&-11&-28\\& & \color{blue}{-25} & & & & \\ \hline &\color{blue}{5}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -25 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrr}-5&5&\color{orangered}{ 19 }&-2&7&-11&-28\\& & \color{orangered}{-25} & & & & \\ \hline &5&\color{orangered}{-6}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&5&19&-2&7&-11&-28\\& & -25& \color{blue}{30} & & & \\ \hline &5&\color{blue}{-6}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 30 } = \color{orangered}{ 28 } $
$$ \begin{array}{c|rrrrrr}-5&5&19&\color{orangered}{ -2 }&7&-11&-28\\& & -25& \color{orangered}{30} & & & \\ \hline &5&-6&\color{orangered}{28}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 28 } = \color{blue}{ -140 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&5&19&-2&7&-11&-28\\& & -25& 30& \color{blue}{-140} & & \\ \hline &5&-6&\color{blue}{28}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -140 \right) } = \color{orangered}{ -133 } $
$$ \begin{array}{c|rrrrrr}-5&5&19&-2&\color{orangered}{ 7 }&-11&-28\\& & -25& 30& \color{orangered}{-140} & & \\ \hline &5&-6&28&\color{orangered}{-133}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -133 \right) } = \color{blue}{ 665 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&5&19&-2&7&-11&-28\\& & -25& 30& -140& \color{blue}{665} & \\ \hline &5&-6&28&\color{blue}{-133}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 665 } = \color{orangered}{ 654 } $
$$ \begin{array}{c|rrrrrr}-5&5&19&-2&7&\color{orangered}{ -11 }&-28\\& & -25& 30& -140& \color{orangered}{665} & \\ \hline &5&-6&28&-133&\color{orangered}{654}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 654 } = \color{blue}{ -3270 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&5&19&-2&7&-11&-28\\& & -25& 30& -140& 665& \color{blue}{-3270} \\ \hline &5&-6&28&-133&\color{blue}{654}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ \left( -3270 \right) } = \color{orangered}{ -3298 } $
$$ \begin{array}{c|rrrrrr}-5&5&19&-2&7&-11&\color{orangered}{ -28 }\\& & -25& 30& -140& 665& \color{orangered}{-3270} \\ \hline &\color{blue}{5}&\color{blue}{-6}&\color{blue}{28}&\color{blue}{-133}&\color{blue}{654}&\color{orangered}{-3298} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{4}-6x^{3}+28x^{2}-133x+654 } $ with a remainder of $ \color{red}{ -3298 } $.