Divide using synthetic division $$ ( 5x^{3}-9x^{2}+28x+6 ) \div ( 5x+1 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&5&-9&28&6\\& & -5& 14& \color{black}{-42} \\ \hline &\color{blue}{5}&\color{blue}{-14}&\color{blue}{42}&\color{orangered}{-36} \end{array} $$The solution is:
$$ \dfrac{ 5x^{3}-9x^{2}+28x+6 }{ x+1 } = \color{blue}{5x^{2}-14x+42} \color{red}{~-~} \dfrac{ \color{red}{ 36 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&5&-9&28&6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 5 }&-9&28&6\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&5&-9&28&6\\& & \color{blue}{-5} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrrr}-1&5&\color{orangered}{ -9 }&28&6\\& & \color{orangered}{-5} & & \\ \hline &5&\color{orangered}{-14}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&5&-9&28&6\\& & -5& \color{blue}{14} & \\ \hline &5&\color{blue}{-14}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 28 } + \color{orangered}{ 14 } = \color{orangered}{ 42 } $
$$ \begin{array}{c|rrrr}-1&5&-9&\color{orangered}{ 28 }&6\\& & -5& \color{orangered}{14} & \\ \hline &5&-14&\color{orangered}{42}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 42 } = \color{blue}{ -42 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&5&-9&28&6\\& & -5& 14& \color{blue}{-42} \\ \hline &5&-14&\color{blue}{42}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -42 \right) } = \color{orangered}{ -36 } $
$$ \begin{array}{c|rrrr}-1&5&-9&28&\color{orangered}{ 6 }\\& & -5& 14& \color{orangered}{-42} \\ \hline &\color{blue}{5}&\color{blue}{-14}&\color{blue}{42}&\color{orangered}{-36} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}-14x+42 } $ with a remainder of $ \color{red}{ -36 } $.