Divide using synthetic division $$ ( 4x^{3}-6x^{2}-14 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&4&-6&0&-14\\& & -24& 180& \color{black}{-1080} \\ \hline &\color{blue}{4}&\color{blue}{-30}&\color{blue}{180}&\color{orangered}{-1094} \end{array} $$The solution is:
$$ \frac{ 4x^{3}-6x^{2}-14 }{ x+6 } = \color{blue}{4x^{2}-30x+180} \color{red}{~-~} \frac{ \color{red}{ 1094 } }{ x+6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&4&-6&0&-14\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 4 }&-6&0&-14\\& & & & \\ \hline &\color{orangered}{4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 4 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&4&-6&0&-14\\& & \color{blue}{-24} & & \\ \hline &\color{blue}{4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -30 } $
$$ \begin{array}{c|rrrr}-6&4&\color{orangered}{ -6 }&0&-14\\& & \color{orangered}{-24} & & \\ \hline &4&\color{orangered}{-30}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -30 \right) } = \color{blue}{ 180 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&4&-6&0&-14\\& & -24& \color{blue}{180} & \\ \hline &4&\color{blue}{-30}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 180 } = \color{orangered}{ 180 } $
$$ \begin{array}{c|rrrr}-6&4&-6&\color{orangered}{ 0 }&-14\\& & -24& \color{orangered}{180} & \\ \hline &4&-30&\color{orangered}{180}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 180 } = \color{blue}{ -1080 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&4&-6&0&-14\\& & -24& 180& \color{blue}{-1080} \\ \hline &4&-30&\color{blue}{180}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ \left( -1080 \right) } = \color{orangered}{ -1094 } $
$$ \begin{array}{c|rrrr}-6&4&-6&0&\color{orangered}{ -14 }\\& & -24& 180& \color{orangered}{-1080} \\ \hline &\color{blue}{4}&\color{blue}{-30}&\color{blue}{180}&\color{orangered}{-1094} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{2}-30x+180 } $ with a remainder of $ \color{red}{ -1094 } $.