Divide using synthetic division $$ ( 6x^{2}+45x-6 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&6&45&-6\\& & -12& \color{black}{-66} \\ \hline &\color{blue}{6}&\color{blue}{33}&\color{orangered}{-72} \end{array} $$The solution is:
$$ \frac{ 6x^{2}+45x-6 }{ x+2 } = \color{blue}{6x+33} \color{red}{~-~} \frac{ \color{red}{ 72 } }{ x+2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-2}&6&45&-6\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 6 }&45&-6\\& & & \\ \hline &\color{orangered}{6}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 6 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&6&45&-6\\& & \color{blue}{-12} & \\ \hline &\color{blue}{6}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 45 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ 33 } $
$$ \begin{array}{c|rrr}-2&6&\color{orangered}{ 45 }&-6\\& & \color{orangered}{-12} & \\ \hline &6&\color{orangered}{33}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 33 } = \color{blue}{ -66 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&6&45&-6\\& & -12& \color{blue}{-66} \\ \hline &6&\color{blue}{33}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -66 \right) } = \color{orangered}{ -72 } $
$$ \begin{array}{c|rrr}-2&6&45&\color{orangered}{ -6 }\\& & -12& \color{orangered}{-66} \\ \hline &\color{blue}{6}&\color{blue}{33}&\color{orangered}{-72} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x+33 } $ with a remainder of $ \color{red}{ -72 } $.