Divide using synthetic division $$ ( x^{2}-3x+7 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&1&-3&7\\& & -5& \color{black}{40} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{orangered}{47} \end{array} $$The solution is:
$$ \frac{ x^{2}-3x+7 }{ x+5 } = \color{blue}{x-8} ~+~ \frac{ \color{red}{ 47 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-5}&1&-3&7\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ 1 }&-3&7\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&1&-3&7\\& & \color{blue}{-5} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrr}-5&1&\color{orangered}{ -3 }&7\\& & \color{orangered}{-5} & \\ \hline &1&\color{orangered}{-8}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 40 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&1&-3&7\\& & -5& \color{blue}{40} \\ \hline &1&\color{blue}{-8}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ 40 } = \color{orangered}{ 47 } $
$$ \begin{array}{c|rrr}-5&1&-3&\color{orangered}{ 7 }\\& & -5& \color{orangered}{40} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{orangered}{47} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x-8 } $ with a remainder of $ \color{red}{ 47 } $.