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