Divide using synthetic division $$ ( 4x^{2}+13x+6 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&4&13&6\\& & -8& \color{black}{-10} \\ \hline &\color{blue}{4}&\color{blue}{5}&\color{orangered}{-4} \end{array} $$The solution is:
$$ \frac{ 4x^{2}+13x+6 }{ x+2 } = \color{blue}{4x+5} \color{red}{~-~} \frac{ \color{red}{ 4 } }{ x+2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-2}&4&13&6\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 4 }&13&6\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 4 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&4&13&6\\& & \color{blue}{-8} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrr}-2&4&\color{orangered}{ 13 }&6\\& & \color{orangered}{-8} & \\ \hline &4&\color{orangered}{5}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&4&13&6\\& & -8& \color{blue}{-10} \\ \hline &4&\color{blue}{5}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrr}-2&4&13&\color{orangered}{ 6 }\\& & -8& \color{orangered}{-10} \\ \hline &\color{blue}{4}&\color{blue}{5}&\color{orangered}{-4} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x+5 } $ with a remainder of $ \color{red}{ -4 } $.