Divide using synthetic division $$ ( 15x^{2}-17x-15 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}1&15&-17&-15\\& & 15& \color{black}{-2} \\ \hline &\color{blue}{15}&\color{blue}{-2}&\color{orangered}{-17} \end{array} $$The solution is:
$$ \frac{ 15x^{2}-17x-15 }{ x-1 } = \color{blue}{15x-2} \color{red}{~-~} \frac{ \color{red}{ 17 } }{ x-1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{1}&15&-17&-15\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}1&\color{orangered}{ 15 }&-17&-15\\& & & \\ \hline &\color{orangered}{15}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 15 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrr}\color{blue}{1}&15&-17&-15\\& & \color{blue}{15} & \\ \hline &\color{blue}{15}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 15 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrr}1&15&\color{orangered}{ -17 }&-15\\& & \color{orangered}{15} & \\ \hline &15&\color{orangered}{-2}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrr}\color{blue}{1}&15&-17&-15\\& & 15& \color{blue}{-2} \\ \hline &15&\color{blue}{-2}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrr}1&15&-17&\color{orangered}{ -15 }\\& & 15& \color{orangered}{-2} \\ \hline &\color{blue}{15}&\color{blue}{-2}&\color{orangered}{-17} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 15x-2 } $ with a remainder of $ \color{red}{ -17 } $.