Divide using synthetic division $$ ( -5x^{3}-34x^{2}-25x+25 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&-5&-34&-25&25\\& & 10& 48& \color{black}{-46} \\ \hline &\color{blue}{-5}&\color{blue}{-24}&\color{blue}{23}&\color{orangered}{-21} \end{array} $$The solution is:
$$ \frac{ -5x^{3}-34x^{2}-25x+25 }{ x+2 } = \color{blue}{-5x^{2}-24x+23} \color{red}{~-~} \frac{ \color{red}{ 21 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-5&-34&-25&25\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ -5 }&-34&-25&25\\& & & & \\ \hline &\color{orangered}{-5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-5&-34&-25&25\\& & \color{blue}{10} & & \\ \hline &\color{blue}{-5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 10 } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrr}-2&-5&\color{orangered}{ -34 }&-25&25\\& & \color{orangered}{10} & & \\ \hline &-5&\color{orangered}{-24}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-5&-34&-25&25\\& & 10& \color{blue}{48} & \\ \hline &-5&\color{blue}{-24}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 48 } = \color{orangered}{ 23 } $
$$ \begin{array}{c|rrrr}-2&-5&-34&\color{orangered}{ -25 }&25\\& & 10& \color{orangered}{48} & \\ \hline &-5&-24&\color{orangered}{23}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 23 } = \color{blue}{ -46 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&-5&-34&-25&25\\& & 10& 48& \color{blue}{-46} \\ \hline &-5&-24&\color{blue}{23}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ \left( -46 \right) } = \color{orangered}{ -21 } $
$$ \begin{array}{c|rrrr}-2&-5&-34&-25&\color{orangered}{ 25 }\\& & 10& 48& \color{orangered}{-46} \\ \hline &\color{blue}{-5}&\color{blue}{-24}&\color{blue}{23}&\color{orangered}{-21} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5x^{2}-24x+23 } $ with a remainder of $ \color{red}{ -21 } $.