Divide using synthetic division $$ ( 7x^{2}-3x+5 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-1&7&-3&5\\& & -7& \color{black}{10} \\ \hline &\color{blue}{7}&\color{blue}{-10}&\color{orangered}{15} \end{array} $$The solution is:
$$ \frac{ 7x^{2}-3x+5 }{ x+1 } = \color{blue}{7x-10} ~+~ \frac{ \color{red}{ 15 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-1}&7&-3&5\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-1&\color{orangered}{ 7 }&-3&5\\& & & \\ \hline &\color{orangered}{7}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 7 } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&7&-3&5\\& & \color{blue}{-7} & \\ \hline &\color{blue}{7}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrr}-1&7&\color{orangered}{ -3 }&5\\& & \color{orangered}{-7} & \\ \hline &7&\color{orangered}{-10}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&7&-3&5\\& & -7& \color{blue}{10} \\ \hline &7&\color{blue}{-10}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 10 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrr}-1&7&-3&\color{orangered}{ 5 }\\& & -7& \color{orangered}{10} \\ \hline &\color{blue}{7}&\color{blue}{-10}&\color{orangered}{15} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 7x-10 } $ with a remainder of $ \color{red}{ 15 } $.