Divide using synthetic division $$ ( 9x^{2}+14x-24 ) \div ( 4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}0&9&14&-24\\& & 0& \color{black}{0} \\ \hline &\color{blue}{9}&\color{blue}{14}&\color{orangered}{-24} \end{array} $$The solution is:
$$ \dfrac{ 9x^{2}+14x-24 }{ x } = \color{blue}{9x+14} \color{red}{~-~} \dfrac{ \color{red}{ 24 } }{ x } $$Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrr}\color{blue}{0}&9&14&-24\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}0&\color{orangered}{ 9 }&14&-24\\& & & \\ \hline &\color{orangered}{9}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 9 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&9&14&-24\\& & \color{blue}{0} & \\ \hline &\color{blue}{9}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ 0 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrr}0&9&\color{orangered}{ 14 }&-24\\& & \color{orangered}{0} & \\ \hline &9&\color{orangered}{14}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 14 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&9&14&-24\\& & 0& \color{blue}{0} \\ \hline &9&\color{blue}{14}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 0 } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrr}0&9&14&\color{orangered}{ -24 }\\& & 0& \color{orangered}{0} \\ \hline &\color{blue}{9}&\color{blue}{14}&\color{orangered}{-24} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x+14 } $ with a remainder of $ \color{red}{ -24 } $.