Divide using synthetic division $$ ( x^{3}+19x^{2}+84x-54 ) \div ( x+10 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-10&1&19&84&-54\\& & -10& -90& \color{black}{60} \\ \hline &\color{blue}{1}&\color{blue}{9}&\color{blue}{-6}&\color{orangered}{6} \end{array} $$The solution is:
$$ \dfrac{ x^{3}+19x^{2}+84x-54 }{ x+10 } = \color{blue}{x^{2}+9x-6} ~+~ \dfrac{ \color{red}{ 6 } }{ x+10 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 10 = 0 $ ( $ x = \color{blue}{ -10 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-10}&1&19&84&-54\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-10&\color{orangered}{ 1 }&19&84&-54\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ 1 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-10}&1&19&84&-54\\& & \color{blue}{-10} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrr}-10&1&\color{orangered}{ 19 }&84&-54\\& & \color{orangered}{-10} & & \\ \hline &1&\color{orangered}{9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ 9 } = \color{blue}{ -90 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-10}&1&19&84&-54\\& & -10& \color{blue}{-90} & \\ \hline &1&\color{blue}{9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 84 } + \color{orangered}{ \left( -90 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}-10&1&19&\color{orangered}{ 84 }&-54\\& & -10& \color{orangered}{-90} & \\ \hline &1&9&\color{orangered}{-6}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-10}&1&19&84&-54\\& & -10& -90& \color{blue}{60} \\ \hline &1&9&\color{blue}{-6}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ 60 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}-10&1&19&84&\color{orangered}{ -54 }\\& & -10& -90& \color{orangered}{60} \\ \hline &\color{blue}{1}&\color{blue}{9}&\color{blue}{-6}&\color{orangered}{6} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+9x-6 } $ with a remainder of $ \color{red}{ 6 } $.