Divide using synthetic division $$ ( 2x^{2}-x-28 ) \div ( 2x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-7&2&-1&-28\\& & -14& \color{black}{105} \\ \hline &\color{blue}{2}&\color{blue}{-15}&\color{orangered}{77} \end{array} $$The solution is:
$$ \frac{ 2x^{2}-x-28 }{ x+7 } = \color{blue}{2x-15} ~+~ \frac{ \color{red}{ 77 } }{ x+7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-7}&2&-1&-28\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-7&\color{orangered}{ 2 }&-1&-28\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 2 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&2&-1&-28\\& & \color{blue}{-14} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrr}-7&2&\color{orangered}{ -1 }&-28\\& & \color{orangered}{-14} & \\ \hline &2&\color{orangered}{-15}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ 105 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&2&-1&-28\\& & -14& \color{blue}{105} \\ \hline &2&\color{blue}{-15}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 105 } = \color{orangered}{ 77 } $
$$ \begin{array}{c|rrr}-7&2&-1&\color{orangered}{ -28 }\\& & -14& \color{orangered}{105} \\ \hline &\color{blue}{2}&\color{blue}{-15}&\color{orangered}{77} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x-15 } $ with a remainder of $ \color{red}{ 77 } $.