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