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