Divide using synthetic division $$ ( 4x^{2}-19x-61 ) \div ( x-7 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}7&4&-19&-61\\& & 28& \color{black}{63} \\ \hline &\color{blue}{4}&\color{blue}{9}&\color{orangered}{2} \end{array} $$The solution is:
$$ \dfrac{ 4x^{2}-19x-61 }{ x-7 } = \color{blue}{4x+9} ~+~ \dfrac{ \color{red}{ 2 } }{ x-7 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{7}&4&-19&-61\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}7&\color{orangered}{ 4 }&-19&-61\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 4 } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&4&-19&-61\\& & \color{blue}{28} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 28 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrr}7&4&\color{orangered}{ -19 }&-61\\& & \color{orangered}{28} & \\ \hline &4&\color{orangered}{9}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 9 } = \color{blue}{ 63 } $.
$$ \begin{array}{c|rrr}\color{blue}{7}&4&-19&-61\\& & 28& \color{blue}{63} \\ \hline &4&\color{blue}{9}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -61 } + \color{orangered}{ 63 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrr}7&4&-19&\color{orangered}{ -61 }\\& & 28& \color{orangered}{63} \\ \hline &\color{blue}{4}&\color{blue}{9}&\color{orangered}{2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x+9 } $ with a remainder of $ \color{red}{ 2 } $.