Divide using synthetic division $$ ( 2x^{2}-17x-38 ) \div ( 2x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-3&2&-17&-38\\& & -6& \color{black}{69} \\ \hline &\color{blue}{2}&\color{blue}{-23}&\color{orangered}{31} \end{array} $$The solution is:
$$ \frac{ 2x^{2}-17x-38 }{ x+3 } = \color{blue}{2x-23} ~+~ \frac{ \color{red}{ 31 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-3}&2&-17&-38\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-3&\color{orangered}{ 2 }&-17&-38\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&2&-17&-38\\& & \color{blue}{-6} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -23 } $
$$ \begin{array}{c|rrr}-3&2&\color{orangered}{ -17 }&-38\\& & \color{orangered}{-6} & \\ \hline &2&\color{orangered}{-23}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -23 \right) } = \color{blue}{ 69 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&2&-17&-38\\& & -6& \color{blue}{69} \\ \hline &2&\color{blue}{-23}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -38 } + \color{orangered}{ 69 } = \color{orangered}{ 31 } $
$$ \begin{array}{c|rrr}-3&2&-17&\color{orangered}{ -38 }\\& & -6& \color{orangered}{69} \\ \hline &\color{blue}{2}&\color{blue}{-23}&\color{orangered}{31} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x-23 } $ with a remainder of $ \color{red}{ 31 } $.