Divide using synthetic division $$ ( 2x^{2}+6x+19 ) \div ( x+7 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}-7&2&6&19\\& & -14& \color{black}{56} \\ \hline &\color{blue}{2}&\color{blue}{-8}&\color{orangered}{75} \end{array} $$The solution is:
$$ \dfrac{ 2x^{2}+6x+19 }{ x+7 } = \color{blue}{2x-8} ~+~ \dfrac{ \color{red}{ 75 } }{ x+7 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-7}&2&6&19\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-7&\color{orangered}{ 2 }&6&19\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 2 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&2&6&19\\& & \color{blue}{-14} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrr}-7&2&\color{orangered}{ 6 }&19\\& & \color{orangered}{-14} & \\ \hline &2&\color{orangered}{-8}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 56 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&2&6&19\\& & -14& \color{blue}{56} \\ \hline &2&\color{blue}{-8}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ 56 } = \color{orangered}{ 75 } $
$$ \begin{array}{c|rrr}-7&2&6&\color{orangered}{ 19 }\\& & -14& \color{orangered}{56} \\ \hline &\color{blue}{2}&\color{blue}{-8}&\color{orangered}{75} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x-8 } $ with a remainder of $ \color{red}{ 75 } $.