Divide using synthetic division $$ ( 5x^{2}-x0 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&5&-1&0\\& & -10& \color{black}{22} \\ \hline &\color{blue}{5}&\color{blue}{-11}&\color{orangered}{22} \end{array} $$The solution is:
$$ \frac{ 5x^{2}-x }{ x+2 } = \color{blue}{5x-11} ~+~ \frac{ \color{red}{ 22 } }{ x+2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-2}&5&-1&0\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 5 }&-1&0\\& & & \\ \hline &\color{orangered}{5}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&5&-1&0\\& & \color{blue}{-10} & \\ \hline &\color{blue}{5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrr}-2&5&\color{orangered}{ -1 }&0\\& & \color{orangered}{-10} & \\ \hline &5&\color{orangered}{-11}& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&5&-1&0\\& & -10& \color{blue}{22} \\ \hline &5&\color{blue}{-11}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 22 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrr}-2&5&-1&\color{orangered}{ 0 }\\& & -10& \color{orangered}{22} \\ \hline &\color{blue}{5}&\color{blue}{-11}&\color{orangered}{22} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x-11 } $ with a remainder of $ \color{red}{ 22 } $.