Divide using synthetic division $$ ( x^{4}-25x^{2}+144 ) \div ( x-4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&1&0&-25&0&144\\& & 4& 16& -36& \color{black}{-144} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{-9}&\color{blue}{-36}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ x^{4}-25x^{2}+144 }{ x-4 } = \color{blue}{x^{3}+4x^{2}-9x-36} $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&0&-25&0&144\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 1 }&0&-25&0&144\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 1 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&0&-25&0&144\\& & \color{blue}{4} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrr}4&1&\color{orangered}{ 0 }&-25&0&144\\& & \color{orangered}{4} & & & \\ \hline &1&\color{orangered}{4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 4 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&0&-25&0&144\\& & 4& \color{blue}{16} & & \\ \hline &1&\color{blue}{4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 16 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}4&1&0&\color{orangered}{ -25 }&0&144\\& & 4& \color{orangered}{16} & & \\ \hline &1&4&\color{orangered}{-9}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&0&-25&0&144\\& & 4& 16& \color{blue}{-36} & \\ \hline &1&4&\color{blue}{-9}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -36 } $
$$ \begin{array}{c|rrrrr}4&1&0&-25&\color{orangered}{ 0 }&144\\& & 4& 16& \color{orangered}{-36} & \\ \hline &1&4&-9&\color{orangered}{-36}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -36 \right) } = \color{blue}{ -144 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&0&-25&0&144\\& & 4& 16& -36& \color{blue}{-144} \\ \hline &1&4&-9&\color{blue}{-36}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 144 } + \color{orangered}{ \left( -144 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}4&1&0&-25&0&\color{orangered}{ 144 }\\& & 4& 16& -36& \color{orangered}{-144} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{-9}&\color{blue}{-36}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+4x^{2}-9x-36 } $ with a remainder of $ \color{red}{ 0 } $.