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