Divide using synthetic division $$ ( -5x^{3}-34x^{2}-25x+75 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&-5&-34&-25&75\\& & 30& 24& \color{black}{6} \\ \hline &\color{blue}{-5}&\color{blue}{-4}&\color{blue}{-1}&\color{orangered}{81} \end{array} $$The solution is:
$$ \frac{ -5x^{3}-34x^{2}-25x+75 }{ x+6 } = \color{blue}{-5x^{2}-4x-1} ~+~ \frac{ \color{red}{ 81 } }{ x+6 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&-5&-34&-25&75\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ -5 }&-34&-25&75\\& & & & \\ \hline &\color{orangered}{-5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&-5&-34&-25&75\\& & \color{blue}{30} & & \\ \hline &\color{blue}{-5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 30 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}-6&-5&\color{orangered}{ -34 }&-25&75\\& & \color{orangered}{30} & & \\ \hline &-5&\color{orangered}{-4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&-5&-34&-25&75\\& & 30& \color{blue}{24} & \\ \hline &-5&\color{blue}{-4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 24 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}-6&-5&-34&\color{orangered}{ -25 }&75\\& & 30& \color{orangered}{24} & \\ \hline &-5&-4&\color{orangered}{-1}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&-5&-34&-25&75\\& & 30& 24& \color{blue}{6} \\ \hline &-5&-4&\color{blue}{-1}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 75 } + \color{orangered}{ 6 } = \color{orangered}{ 81 } $
$$ \begin{array}{c|rrrr}-6&-5&-34&-25&\color{orangered}{ 75 }\\& & 30& 24& \color{orangered}{6} \\ \hline &\color{blue}{-5}&\color{blue}{-4}&\color{blue}{-1}&\color{orangered}{81} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5x^{2}-4x-1 } $ with a remainder of $ \color{red}{ 81 } $.