Divide using synthetic division $$ ( -6x^{3}-5x^{2}+18x+11 ) \div ( 3x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&-6&-5&18&11\\& & -12& -34& \color{black}{-32} \\ \hline &\color{blue}{-6}&\color{blue}{-17}&\color{blue}{-16}&\color{orangered}{-21} \end{array} $$The solution is:
$$ \frac{ -6x^{3}-5x^{2}+18x+11 }{ x-2 } = \color{blue}{-6x^{2}-17x-16} \color{red}{~-~} \frac{ \color{red}{ 21 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-6&-5&18&11\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ -6 }&-5&18&11\\& & & & \\ \hline &\color{orangered}{-6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-6&-5&18&11\\& & \color{blue}{-12} & & \\ \hline &\color{blue}{-6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrr}2&-6&\color{orangered}{ -5 }&18&11\\& & \color{orangered}{-12} & & \\ \hline &-6&\color{orangered}{-17}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ -34 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-6&-5&18&11\\& & -12& \color{blue}{-34} & \\ \hline &-6&\color{blue}{-17}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -34 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrr}2&-6&-5&\color{orangered}{ 18 }&11\\& & -12& \color{orangered}{-34} & \\ \hline &-6&-17&\color{orangered}{-16}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-6&-5&18&11\\& & -12& -34& \color{blue}{-32} \\ \hline &-6&-17&\color{blue}{-16}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -21 } $
$$ \begin{array}{c|rrrr}2&-6&-5&18&\color{orangered}{ 11 }\\& & -12& -34& \color{orangered}{-32} \\ \hline &\color{blue}{-6}&\color{blue}{-17}&\color{blue}{-16}&\color{orangered}{-21} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -6x^{2}-17x-16 } $ with a remainder of $ \color{red}{ -21 } $.