Divide using synthetic division $$ ( -10x^{2}+31x-6 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}6&-10&31&-6\\& & -60& \color{black}{-174} \\ \hline &\color{blue}{-10}&\color{blue}{-29}&\color{orangered}{-180} \end{array} $$The solution is:
$$ \frac{ -10x^{2}+31x-6 }{ x-6 } = \color{blue}{-10x-29} \color{red}{~-~} \frac{ \color{red}{ 180 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{6}&-10&31&-6\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}6&\color{orangered}{ -10 }&31&-6\\& & & \\ \hline &\color{orangered}{-10}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ -60 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&-10&31&-6\\& & \color{blue}{-60} & \\ \hline &\color{blue}{-10}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 31 } + \color{orangered}{ \left( -60 \right) } = \color{orangered}{ -29 } $
$$ \begin{array}{c|rrr}6&-10&\color{orangered}{ 31 }&-6\\& & \color{orangered}{-60} & \\ \hline &-10&\color{orangered}{-29}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -29 \right) } = \color{blue}{ -174 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&-10&31&-6\\& & -60& \color{blue}{-174} \\ \hline &-10&\color{blue}{-29}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -174 \right) } = \color{orangered}{ -180 } $
$$ \begin{array}{c|rrr}6&-10&31&\color{orangered}{ -6 }\\& & -60& \color{orangered}{-174} \\ \hline &\color{blue}{-10}&\color{blue}{-29}&\color{orangered}{-180} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -10x-29 } $ with a remainder of $ \color{red}{ -180 } $.