Divide using synthetic division $$ ( -5x^{2}-x-10 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}2&-5&-1&-10\\& & -10& \color{black}{-22} \\ \hline &\color{blue}{-5}&\color{blue}{-11}&\color{orangered}{-32} \end{array} $$The solution is:
$$ \frac{ -5x^{2}-x-10 }{ x-2 } = \color{blue}{-5x-11} \color{red}{~-~} \frac{ \color{red}{ 32 } }{ x-2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{2}&-5&-1&-10\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}2&\color{orangered}{ -5 }&-1&-10\\& & & \\ \hline &\color{orangered}{-5}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&-5&-1&-10\\& & \color{blue}{-10} & \\ \hline &\color{blue}{-5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrr}2&-5&\color{orangered}{ -1 }&-10\\& & \color{orangered}{-10} & \\ \hline &-5&\color{orangered}{-11}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ -22 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&-5&-1&-10\\& & -10& \color{blue}{-22} \\ \hline &-5&\color{blue}{-11}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -22 \right) } = \color{orangered}{ -32 } $
$$ \begin{array}{c|rrr}2&-5&-1&\color{orangered}{ -10 }\\& & -10& \color{orangered}{-22} \\ \hline &\color{blue}{-5}&\color{blue}{-11}&\color{orangered}{-32} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5x-11 } $ with a remainder of $ \color{red}{ -32 } $.