Divide using synthetic division $$ ( 4x^{4}+25x^{3}+2x^{2}-25x-8 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-6&4&25&2&-25&-8\\& & -24& -6& 24& \color{black}{6} \\ \hline &\color{blue}{4}&\color{blue}{1}&\color{blue}{-4}&\color{blue}{-1}&\color{orangered}{-2} \end{array} $$The solution is:
$$ \frac{ 4x^{4}+25x^{3}+2x^{2}-25x-8 }{ x+6 } = \color{blue}{4x^{3}+x^{2}-4x-1} \color{red}{~-~} \frac{ \color{red}{ 2 } }{ x+6 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&4&25&2&-25&-8\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-6&\color{orangered}{ 4 }&25&2&-25&-8\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&4&25&2&-25&-8\\& & \color{blue}{-24} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrr}-6&4&\color{orangered}{ 25 }&2&-25&-8\\& & \color{orangered}{-24} & & & \\ \hline &4&\color{orangered}{1}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 1 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&4&25&2&-25&-8\\& & -24& \color{blue}{-6} & & \\ \hline &4&\color{blue}{1}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}-6&4&25&\color{orangered}{ 2 }&-25&-8\\& & -24& \color{orangered}{-6} & & \\ \hline &4&1&\color{orangered}{-4}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&4&25&2&-25&-8\\& & -24& -6& \color{blue}{24} & \\ \hline &4&1&\color{blue}{-4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 24 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}-6&4&25&2&\color{orangered}{ -25 }&-8\\& & -24& -6& \color{orangered}{24} & \\ \hline &4&1&-4&\color{orangered}{-1}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&4&25&2&-25&-8\\& & -24& -6& 24& \color{blue}{6} \\ \hline &4&1&-4&\color{blue}{-1}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 6 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-6&4&25&2&-25&\color{orangered}{ -8 }\\& & -24& -6& 24& \color{orangered}{6} \\ \hline &\color{blue}{4}&\color{blue}{1}&\color{blue}{-4}&\color{blue}{-1}&\color{orangered}{-2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}+x^{2}-4x-1 } $ with a remainder of $ \color{red}{ -2 } $.