Divide using synthetic division $$ ( 4x^{4}-10x^{2}-24 ) \div ( 2x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&4&0&-10&0&-24\\& & 16& 64& 216& \color{black}{864} \\ \hline &\color{blue}{4}&\color{blue}{16}&\color{blue}{54}&\color{blue}{216}&\color{orangered}{840} \end{array} $$The solution is:
$$ \frac{ 4x^{4}-10x^{2}-24 }{ x-4 } = \color{blue}{4x^{3}+16x^{2}+54x+216} ~+~ \frac{ \color{red}{ 840 } }{ x-4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&4&0&-10&0&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 4 }&0&-10&0&-24\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 4 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&4&0&-10&0&-24\\& & \color{blue}{16} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 16 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrr}4&4&\color{orangered}{ 0 }&-10&0&-24\\& & \color{orangered}{16} & & & \\ \hline &4&\color{orangered}{16}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 16 } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&4&0&-10&0&-24\\& & 16& \color{blue}{64} & & \\ \hline &4&\color{blue}{16}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 64 } = \color{orangered}{ 54 } $
$$ \begin{array}{c|rrrrr}4&4&0&\color{orangered}{ -10 }&0&-24\\& & 16& \color{orangered}{64} & & \\ \hline &4&16&\color{orangered}{54}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 54 } = \color{blue}{ 216 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&4&0&-10&0&-24\\& & 16& 64& \color{blue}{216} & \\ \hline &4&16&\color{blue}{54}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 216 } = \color{orangered}{ 216 } $
$$ \begin{array}{c|rrrrr}4&4&0&-10&\color{orangered}{ 0 }&-24\\& & 16& 64& \color{orangered}{216} & \\ \hline &4&16&54&\color{orangered}{216}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 216 } = \color{blue}{ 864 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&4&0&-10&0&-24\\& & 16& 64& 216& \color{blue}{864} \\ \hline &4&16&54&\color{blue}{216}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 864 } = \color{orangered}{ 840 } $
$$ \begin{array}{c|rrrrr}4&4&0&-10&0&\color{orangered}{ -24 }\\& & 16& 64& 216& \color{orangered}{864} \\ \hline &\color{blue}{4}&\color{blue}{16}&\color{blue}{54}&\color{blue}{216}&\color{orangered}{840} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}+16x^{2}+54x+216 } $ with a remainder of $ \color{red}{ 840 } $.