Divide using synthetic division $$ ( 3x^{3}+5x^{2}-4x-6 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&3&5&-4&-6\\& & 6& 22& \color{black}{36} \\ \hline &\color{blue}{3}&\color{blue}{11}&\color{blue}{18}&\color{orangered}{30} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+5x^{2}-4x-6 }{ x-2 } = \color{blue}{3x^{2}+11x+18} ~+~ \frac{ \color{red}{ 30 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&5&-4&-6\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 3 }&5&-4&-6\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 3 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&5&-4&-6\\& & \color{blue}{6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 6 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrr}2&3&\color{orangered}{ 5 }&-4&-6\\& & \color{orangered}{6} & & \\ \hline &3&\color{orangered}{11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 11 } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&5&-4&-6\\& & 6& \color{blue}{22} & \\ \hline &3&\color{blue}{11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 22 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}2&3&5&\color{orangered}{ -4 }&-6\\& & 6& \color{orangered}{22} & \\ \hline &3&11&\color{orangered}{18}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 18 } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&3&5&-4&-6\\& & 6& 22& \color{blue}{36} \\ \hline &3&11&\color{blue}{18}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 36 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrrr}2&3&5&-4&\color{orangered}{ -6 }\\& & 6& 22& \color{orangered}{36} \\ \hline &\color{blue}{3}&\color{blue}{11}&\color{blue}{18}&\color{orangered}{30} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}+11x+18 } $ with a remainder of $ \color{red}{ 30 } $.