Divide using synthetic division $$ ( 2x^{2}+4x-18 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-5&2&4&-18\\& & -10& \color{black}{30} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{orangered}{12} \end{array} $$The solution is:
$$ \frac{ 2x^{2}+4x-18 }{ x+5 } = \color{blue}{2x-6} ~+~ \frac{ \color{red}{ 12 } }{ x+5 } $$Explanation
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.
$$ \begin{array}{c|rrr}\color{blue}{-5}&2&4&-18\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-5&\color{orangered}{ 2 }&4&-18\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 2 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&2&4&-18\\& & \color{blue}{-10} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrr}-5&2&\color{orangered}{ 4 }&-18\\& & \color{orangered}{-10} & \\ \hline &2&\color{orangered}{-6}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrr}\color{blue}{-5}&2&4&-18\\& & -10& \color{blue}{30} \\ \hline &2&\color{blue}{-6}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 30 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrr}-5&2&4&\color{orangered}{ -18 }\\& & -10& \color{orangered}{30} \\ \hline &\color{blue}{2}&\color{blue}{-6}&\color{orangered}{12} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x-6 } $ with a remainder of $ \color{red}{ 12 } $.