Divide using synthetic division $$ ( x^{4}+13x^{2}-48 ) \div ( -4x ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&1&0&13&0&-48\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{13}&\color{blue}{0}&\color{orangered}{-48} \end{array} $$The solution is:
$$ \dfrac{ x^{4}+13x^{2}-48 }{ x } = \color{blue}{x^{3}+13x} \color{red}{~-~} \dfrac{ \color{red}{ 48 } }{ x } $$Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&0&13&0&-48\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 1 }&0&13&0&-48\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 1 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&0&13&0&-48\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}0&1&\color{orangered}{ 0 }&13&0&-48\\& & \color{orangered}{0} & & & \\ \hline &1&\color{orangered}{0}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&0&13&0&-48\\& & 0& \color{blue}{0} & & \\ \hline &1&\color{blue}{0}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ 0 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrrr}0&1&0&\color{orangered}{ 13 }&0&-48\\& & 0& \color{orangered}{0} & & \\ \hline &1&0&\color{orangered}{13}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 13 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&0&13&0&-48\\& & 0& 0& \color{blue}{0} & \\ \hline &1&0&\color{blue}{13}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}0&1&0&13&\color{orangered}{ 0 }&-48\\& & 0& 0& \color{orangered}{0} & \\ \hline &1&0&13&\color{orangered}{0}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&1&0&13&0&-48\\& & 0& 0& 0& \color{blue}{0} \\ \hline &1&0&13&\color{blue}{0}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -48 } + \color{orangered}{ 0 } = \color{orangered}{ -48 } $
$$ \begin{array}{c|rrrrr}0&1&0&13&0&\color{orangered}{ -48 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{13}&\color{blue}{0}&\color{orangered}{-48} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+13x } $ with a remainder of $ \color{red}{ -48 } $.