Divide using synthetic division $$ ( 4x^{5}+24x^{4}+18x^{3}-10x^{2}+2 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-5&4&24&18&-10&0&2\\& & -20& -20& 10& 0& \color{black}{0} \\ \hline &\color{blue}{4}&\color{blue}{4}&\color{blue}{-2}&\color{blue}{0}&\color{blue}{0}&\color{orangered}{2} \end{array} $$The solution is:
$$ \frac{ 4x^{5}+24x^{4}+18x^{3}-10x^{2}+2 }{ x+5 } = \color{blue}{4x^{4}+4x^{3}-2x^{2}} ~+~ \frac{ \color{red}{ 2 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&4&24&18&-10&0&2\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-5&\color{orangered}{ 4 }&24&18&-10&0&2\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 4 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&4&24&18&-10&0&2\\& & \color{blue}{-20} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrr}-5&4&\color{orangered}{ 24 }&18&-10&0&2\\& & \color{orangered}{-20} & & & & \\ \hline &4&\color{orangered}{4}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 4 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&4&24&18&-10&0&2\\& & -20& \color{blue}{-20} & & & \\ \hline &4&\color{blue}{4}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrr}-5&4&24&\color{orangered}{ 18 }&-10&0&2\\& & -20& \color{orangered}{-20} & & & \\ \hline &4&4&\color{orangered}{-2}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&4&24&18&-10&0&2\\& & -20& -20& \color{blue}{10} & & \\ \hline &4&4&\color{blue}{-2}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 10 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}-5&4&24&18&\color{orangered}{ -10 }&0&2\\& & -20& -20& \color{orangered}{10} & & \\ \hline &4&4&-2&\color{orangered}{0}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&4&24&18&-10&0&2\\& & -20& -20& 10& \color{blue}{0} & \\ \hline &4&4&-2&\color{blue}{0}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}-5&4&24&18&-10&\color{orangered}{ 0 }&2\\& & -20& -20& 10& \color{orangered}{0} & \\ \hline &4&4&-2&0&\color{orangered}{0}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&4&24&18&-10&0&2\\& & -20& -20& 10& 0& \color{blue}{0} \\ \hline &4&4&-2&0&\color{blue}{0}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrr}-5&4&24&18&-10&0&\color{orangered}{ 2 }\\& & -20& -20& 10& 0& \color{orangered}{0} \\ \hline &\color{blue}{4}&\color{blue}{4}&\color{blue}{-2}&\color{blue}{0}&\color{blue}{0}&\color{orangered}{2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{4}+4x^{3}-2x^{2} } $ with a remainder of $ \color{red}{ 2 } $.