Divide using synthetic division $$ ( -10x^{2}+18x+13 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}6&-10&18&13\\& & -60& \color{black}{-252} \\ \hline &\color{blue}{-10}&\color{blue}{-42}&\color{orangered}{-239} \end{array} $$The solution is:
$$ \frac{ -10x^{2}+18x+13 }{ x-6 } = \color{blue}{-10x-42} \color{red}{~-~} \frac{ \color{red}{ 239 } }{ x-6 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{6}&-10&18&13\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}6&\color{orangered}{ -10 }&18&13\\& & & \\ \hline &\color{orangered}{-10}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&-10&18&13\\& & \color{blue}{-60} & \\ \hline &\color{blue}{-10}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -60 \right) } = \color{orangered}{ -42 } $
$$ \begin{array}{c|rrr}6&-10&\color{orangered}{ 18 }&13\\& & \color{orangered}{-60} & \\ \hline &-10&\color{orangered}{-42}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -42 \right) } = \color{blue}{ -252 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&-10&18&13\\& & -60& \color{blue}{-252} \\ \hline &-10&\color{blue}{-42}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -252 \right) } = \color{orangered}{ -239 } $
$$ \begin{array}{c|rrr}6&-10&18&\color{orangered}{ 13 }\\& & -60& \color{orangered}{-252} \\ \hline &\color{blue}{-10}&\color{blue}{-42}&\color{orangered}{-239} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -10x-42 } $ with a remainder of $ \color{red}{ -239 } $.