Divide using synthetic division $$ ( 7x^{2}-38x+27 ) \div ( x-3 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}3&7&-38&27\\& & 21& \color{black}{-51} \\ \hline &\color{blue}{7}&\color{blue}{-17}&\color{orangered}{-24} \end{array} $$The solution is:
$$ \dfrac{ 7x^{2}-38x+27 }{ x-3 } = \color{blue}{7x-17} \color{red}{~-~} \dfrac{ \color{red}{ 24 } }{ x-3 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{3}&7&-38&27\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ 7 }&-38&27\\& & & \\ \hline &\color{orangered}{7}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 7 } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&7&-38&27\\& & \color{blue}{21} & \\ \hline &\color{blue}{7}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -38 } + \color{orangered}{ 21 } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrr}3&7&\color{orangered}{ -38 }&27\\& & \color{orangered}{21} & \\ \hline &7&\color{orangered}{-17}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ -51 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&7&-38&27\\& & 21& \color{blue}{-51} \\ \hline &7&\color{blue}{-17}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 27 } + \color{orangered}{ \left( -51 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrr}3&7&-38&\color{orangered}{ 27 }\\& & 21& \color{orangered}{-51} \\ \hline &\color{blue}{7}&\color{blue}{-17}&\color{orangered}{-24} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 7x-17 } $ with a remainder of $ \color{red}{ -24 } $.