Divide using synthetic division $$ ( 2x^{2}-39x+107 ) \div ( 5x+15 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-15&2&-39&107\\& & -30& \color{black}{1035} \\ \hline &\color{blue}{2}&\color{blue}{-69}&\color{orangered}{1142} \end{array} $$The solution is:
$$ \frac{ 2x^{2}-39x+107 }{ x+15 } = \color{blue}{2x-69} ~+~ \frac{ \color{red}{ 1142 } }{ x+15 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 15 = 0 $ ( $ x = \color{blue}{ -15 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-15}&2&-39&107\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-15&\color{orangered}{ 2 }&-39&107\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -15 } \cdot \color{blue}{ 2 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrr}\color{blue}{-15}&2&-39&107\\& & \color{blue}{-30} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -39 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -69 } $
$$ \begin{array}{c|rrr}-15&2&\color{orangered}{ -39 }&107\\& & \color{orangered}{-30} & \\ \hline &2&\color{orangered}{-69}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -15 } \cdot \color{blue}{ \left( -69 \right) } = \color{blue}{ 1035 } $.
$$ \begin{array}{c|rrr}\color{blue}{-15}&2&-39&107\\& & -30& \color{blue}{1035} \\ \hline &2&\color{blue}{-69}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 107 } + \color{orangered}{ 1035 } = \color{orangered}{ 1142 } $
$$ \begin{array}{c|rrr}-15&2&-39&\color{orangered}{ 107 }\\& & -30& \color{orangered}{1035} \\ \hline &\color{blue}{2}&\color{blue}{-69}&\color{orangered}{1142} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x-69 } $ with a remainder of $ \color{red}{ 1142 } $.