Divide using synthetic division $$ ( -17x^{2}+15x-25 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}5&-17&15&-25\\& & -85& \color{black}{-350} \\ \hline &\color{blue}{-17}&\color{blue}{-70}&\color{orangered}{-375} \end{array} $$The solution is:
$$ \frac{ -17x^{2}+15x-25 }{ x-5 } = \color{blue}{-17x-70} \color{red}{~-~} \frac{ \color{red}{ 375 } }{ x-5 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{5}&-17&15&-25\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}5&\color{orangered}{ -17 }&15&-25\\& & & \\ \hline &\color{orangered}{-17}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ -85 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&-17&15&-25\\& & \color{blue}{-85} & \\ \hline &\color{blue}{-17}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -85 \right) } = \color{orangered}{ -70 } $
$$ \begin{array}{c|rrr}5&-17&\color{orangered}{ 15 }&-25\\& & \color{orangered}{-85} & \\ \hline &-17&\color{orangered}{-70}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -70 \right) } = \color{blue}{ -350 } $.
$$ \begin{array}{c|rrr}\color{blue}{5}&-17&15&-25\\& & -85& \color{blue}{-350} \\ \hline &-17&\color{blue}{-70}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ \left( -350 \right) } = \color{orangered}{ -375 } $
$$ \begin{array}{c|rrr}5&-17&15&\color{orangered}{ -25 }\\& & -85& \color{orangered}{-350} \\ \hline &\color{blue}{-17}&\color{blue}{-70}&\color{orangered}{-375} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -17x-70 } $ with a remainder of $ \color{red}{ -375 } $.