Divide using synthetic division $$ ( 10x^{2}+15x+20 ) \div ( x+5 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&10&15&20\\& & -50& \color{black}{175} \\ \hline &\color{blue}{10}&\color{blue}{-35}&\color{orangered}{195} \end{array} $$The solution is:
$$ \dfrac{ 10x^{2}+15x+20 }{ x+5 } = \color{blue}{10x-35} ~+~ \dfrac{ \color{red}{ 195 } }{ x+5 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-5}&10&15&20\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ 10 }&15&20\\& & & \\ \hline &\color{orangered}{10}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 10 } = \color{blue}{ -50 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&10&15&20\\& & \color{blue}{-50} & \\ \hline &\color{blue}{10}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -50 \right) } = \color{orangered}{ -35 } $
$$ \begin{array}{c|rrr}-5&10&\color{orangered}{ 15 }&20\\& & \color{orangered}{-50} & \\ \hline &10&\color{orangered}{-35}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -35 \right) } = \color{blue}{ 175 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&10&15&20\\& & -50& \color{blue}{175} \\ \hline &10&\color{blue}{-35}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ 175 } = \color{orangered}{ 195 } $
$$ \begin{array}{c|rrr}-5&10&15&\color{orangered}{ 20 }\\& & -50& \color{orangered}{175} \\ \hline &\color{blue}{10}&\color{blue}{-35}&\color{orangered}{195} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 10x-35 } $ with a remainder of $ \color{red}{ 195 } $.