Divide using synthetic division $$ ( 7x^{2}+11x+13 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-1&7&11&13\\& & -7& \color{black}{-4} \\ \hline &\color{blue}{7}&\color{blue}{4}&\color{orangered}{9} \end{array} $$The solution is:
$$ \frac{ 7x^{2}+11x+13 }{ x+1 } = \color{blue}{7x+4} ~+~ \frac{ \color{red}{ 9 } }{ x+1 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-1}&7&11&13\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-1&\color{orangered}{ 7 }&11&13\\& & & \\ \hline &\color{orangered}{7}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&7&11&13\\& & \color{blue}{-7} & \\ \hline &\color{blue}{7}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrr}-1&7&\color{orangered}{ 11 }&13\\& & \color{orangered}{-7} & \\ \hline &7&\color{orangered}{4}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 4 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&7&11&13\\& & -7& \color{blue}{-4} \\ \hline &7&\color{blue}{4}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrr}-1&7&11&\color{orangered}{ 13 }\\& & -7& \color{orangered}{-4} \\ \hline &\color{blue}{7}&\color{blue}{4}&\color{orangered}{9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 7x+4 } $ with a remainder of $ \color{red}{ 9 } $.