Divide using synthetic division $$ ( x^{2}-7x-5 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}2&1&-7&-5\\& & 2& \color{black}{-10} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{orangered}{-15} \end{array} $$The solution is:
$$ \frac{ x^{2}-7x-5 }{ x-2 } = \color{blue}{x-5} \color{red}{~-~} \frac{ \color{red}{ 15 } }{ x-2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{2}&1&-7&-5\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}2&\color{orangered}{ 1 }&-7&-5\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&1&-7&-5\\& & \color{blue}{2} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 2 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrr}2&1&\color{orangered}{ -7 }&-5\\& & \color{orangered}{2} & \\ \hline &1&\color{orangered}{-5}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&1&-7&-5\\& & 2& \color{blue}{-10} \\ \hline &1&\color{blue}{-5}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrr}2&1&-7&\color{orangered}{ -5 }\\& & 2& \color{orangered}{-10} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{orangered}{-15} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x-5 } $ with a remainder of $ \color{red}{ -15 } $.