Divide using synthetic division $$ ( x^{4}+2x^{3}-13x^{2}-38x-24 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&1&2&-13&-38&-24\\& & 4& 24& 44& \color{black}{24} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{11}&\color{blue}{6}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}+2x^{3}-13x^{2}-38x-24 }{ x-4 } = \color{blue}{x^{3}+6x^{2}+11x+6} $$Explanation
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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&2&-13&-38&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 1 }&2&-13&-38&-24\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&2&-13&-38&-24\\& & \color{blue}{4} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 4 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrr}4&1&\color{orangered}{ 2 }&-13&-38&-24\\& & \color{orangered}{4} & & & \\ \hline &1&\color{orangered}{6}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 6 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&2&-13&-38&-24\\& & 4& \color{blue}{24} & & \\ \hline &1&\color{blue}{6}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 24 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrrr}4&1&2&\color{orangered}{ -13 }&-38&-24\\& & 4& \color{orangered}{24} & & \\ \hline &1&6&\color{orangered}{11}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 11 } = \color{blue}{ 44 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&2&-13&-38&-24\\& & 4& 24& \color{blue}{44} & \\ \hline &1&6&\color{blue}{11}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -38 } + \color{orangered}{ 44 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrr}4&1&2&-13&\color{orangered}{ -38 }&-24\\& & 4& 24& \color{orangered}{44} & \\ \hline &1&6&11&\color{orangered}{6}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 6 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&1&2&-13&-38&-24\\& & 4& 24& 44& \color{blue}{24} \\ \hline &1&6&11&\color{blue}{6}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 24 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}4&1&2&-13&-38&\color{orangered}{ -24 }\\& & 4& 24& 44& \color{orangered}{24} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{11}&\color{blue}{6}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+6x^{2}+11x+6 } $ with a remainder of $ \color{red}{ 0 } $.