Divide using synthetic division $$ ( -25x^{2}-40x-17 ) \div ( 5x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&-25&-40&-17\\& & 125& \color{black}{-425} \\ \hline &\color{blue}{-25}&\color{blue}{85}&\color{orangered}{-442} \end{array} $$The solution is:
$$ \frac{ -25x^{2}-40x-17 }{ x+5 } = \color{blue}{-25x+85} \color{red}{~-~} \frac{ \color{red}{ 442 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-5}&-25&-40&-17\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ -25 }&-40&-17\\& & & \\ \hline &\color{orangered}{-25}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -25 \right) } = \color{blue}{ 125 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&-25&-40&-17\\& & \color{blue}{125} & \\ \hline &\color{blue}{-25}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -40 } + \color{orangered}{ 125 } = \color{orangered}{ 85 } $
$$ \begin{array}{c|rrr}-5&-25&\color{orangered}{ -40 }&-17\\& & \color{orangered}{125} & \\ \hline &-25&\color{orangered}{85}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 85 } = \color{blue}{ -425 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&-25&-40&-17\\& & 125& \color{blue}{-425} \\ \hline &-25&\color{blue}{85}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ \left( -425 \right) } = \color{orangered}{ -442 } $
$$ \begin{array}{c|rrr}-5&-25&-40&\color{orangered}{ -17 }\\& & 125& \color{orangered}{-425} \\ \hline &\color{blue}{-25}&\color{blue}{85}&\color{orangered}{-442} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -25x+85 } $ with a remainder of $ \color{red}{ -442 } $.